Mechanics and energy absorption of a functionally graded cylinder subjected to axial loading

International Journal of Engineering Science 78 (2014) 18–26

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Mechanics and energy absorption of a functionally graded cylinder subjected to axial loading Victor Birman ⇑ Engineering Education Center, Missouri University of Science and Technology, 12837 Flushing Meadows Drive, Suite 210, St. Louis, MO 63131, USA

a r t i c l e

i n f o

Article history: Received 28 November 2013 Received in revised form 4 January 2014 Accepted 7 January 2014 Available online 14 February 2014 Keywords: Functionally graded material Cylindrical shell Energy absorption Foams

a b s t r a c t Mechanics of a functionally graded cylinder subject to static or dynamic axial loading is considered including a potential application as energy absorber. The grading in the radial direction is such that the mass density and stiffness are power functions of the radial coordinate as may be the case with variable-density open-cell or closed-cell foams. Exact solutions are obtained in the static problem and in the case where the applied load is a periodic function of time. The absorption of energy is analyzed in the static problem that is reduced to an easily implementable nondimensional formulation. It is demonstrated that by grading the material in the radial direction it is possible to achieve higher energy absorption while at the same time saving weight. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Functionally graded materials have been analyzed in a variety of applications for the last three decades (Birman & Byrd, 2007; Birman, Keil, & Hosder, 2012; Miyamoto, 1999; Paulino, 2006; Suresh & Mortensen, 1998). The pioneering work on the stress problem in cylinders with longitudinally and radially varying properties was published by Lekhnitskii (1963). Functional grading of material can be used to achieve a variety of goals, including alleviation of residual stresses, reducing stresses during lifetime of the structure, improvement of stability and dynamic response, preventing fracture and fatigue, etc. One of intriguing possibilities is related to using low-stiffness regions of a graded structure for enhanced energy absorption. Such attempt is motivated by the observation that materials with a reduced stiffness possess higher toughness. This phenomenon has been documented for both engineering and biological materials (e.g., Ashby, Gibson, Wegst, & Olive, 1995; Fratzl, Gupta, Paschalis, & Roschger, 2004). However, there are limits to a possible stiffness reduction aimed at achieving higher resilience and toughness superimposed by strength requirements. Accordingly, the problem of energy absorption and toughness of functionally graded materials should be considered in conjunction with the stress problem including the distribution of local stresses and the check of local strength. The problem of energy absorption by engineering materials has been considered in numerous studies. Examples are found in recent reviews, such as a comparison of various energy absorbing materials (Qiao, Yang, & Bobaru, 2008), as well as research of peculiar energy absorption mechanisms and effects in particular material systems including nanocomposites (Sun, Gibson, Gordaninejad, & Suhr, 2009), various foams (Chakravarty, 2010), composite materials under axial loading (Lau, Said, & Yaakob, 2012), and ballistic helmets (Kulkarni, Gao, Horner, Zheng, & David, 2013). Foams represent an example of a highly efficient energy absorbing material. An example of recent foam application concepts aimed at maximum energy absorption is its use in personal armor. The stiffness and energy absorption capacity of ⇑ Tel.: +1 314 835 9818. E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijengsci.2014.01.002 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.

V. Birman / International Journal of Engineering Science 78 (2014) 18–26

19

foam depend on its density and architecture (open-cell vs. closed-cell foams). The beneficial effect of a closed-cell polyurethane foam layer on the blast response of sandwich panels was investigated (Bahei-El-Din & Dvorak, 2007, 2008). The designs considered in this research utilized a solid polyurethane layer and a polyurethane foam layer between the outer facing and core. The impact energy being partially absorbed by these layers, the authors achieved a reduction in the peak kinetic energy and core compression by almost 50%. The study utilizing a similar approach with either solid polyurethane or polyurethane foam interlayer between the outer facing and core demonstrated their effectiveness in the mitigation of the effect of a low velocity impact (Dvorak & Suvorov, 2006; Suvorov & Dvorak, 2005). In general, foams present a designer with the desirable flexibility enabling him to utilize spatial variations of the density, strength and stiffness, tailoring the response of the structure to specific design requirements. Besides the previously mentioned work of Lekhnitskii (1963), functionally graded cylinders under axial loading have been considered in both static and dynamic applications (e.g., Darabi, Darvizeh, & Darvizeh, 2008; Liew, Zhao, & Lee, 2012). In these studies the cylinders were characterized by various theories of shells downplaying the three-dimensional nature of the problem. Contrary to the efforts utilizing shell theories, the present problem is solved using a three-dimensional formulation of the theory of elasticity, expanding the approach of Lekhnitskii without limitations on the thickness of the cylinder. In addition to the solution of the static and dynamic stress problems, the paper demonstrates the absorption of energy under static compression facilitated by grading the material density in the radial direction. Note that an attempt to maximize the energy absorption by using a functionally graded material with a strategically reduced local stiffness can be traced to biology. For example, the biological enthesis (tendon-to-bone insertion site) is characterized by functional grading and a dip in the stiffness between tendon and bone (Thomopoulos, Williams, Gimbel, Favata, & Soslowsky, 2003). The research of the enthesis may lead to the assumption that such distribution of properties is conducive to higher resilience of the enthesis (e.g., Genin et al., 2009; Liu, Thomopoulos, Birman, Li, & Genin, 2012). While contrary to the case of enthesis in the problem considered in the paper the material is graded in the direction perpendicular to the load rather than along the load, the analogy with natural biological structures deserves an acknowledgement. 2. Analysis Consider a cylindrical shell of the inner and outer radii r = a and r = b, respectively, and the length L that is supported at one end and loaded by an axial static or dynamic force at the opposite end (Fig. 1). The loaded end of the shell (z = L) is not restrained. The supported end (z = 0) is prevented from axial displacements but free to expand in the radial direction, so that it does not constrain radial deformations of the shell. The shell is manufactured from a quasi-isotropic material that is functionally graded in the radial direction so that the properties, including the stress tensor, vary with the radial coordinate, but the material remains locally isotropic at every point. This implies grading on the macroscopic scale where the mass density is a slowly changing function of the radial coordinate. The analysis is conducted by the following assumptions: 1. The state of generalized plane strain prevails throughout the structure, i.e. each cross section perpendicular to the axis of the cylinder remains plane during deformation, without warping; 2. The problem is both physically and geometrically linear. 3. The cylinder remains stable, i.e. we concentrate on the stress and energy absorption analyses, without concerns related to structural stability. The second assumption implies that even for low-density sections of the material, deformations are small and the stress–strain relationships are linear elastic. In the following, we consider an example of open-cell foam that possesses a nonlinear response in the large strain range that is typical for some of energy absorption applications. However, if this material is employed in the structure experiencing multiple repeated or periodic loads, the nonlinear part of the response

r=a

z P

r=b r L Fig. 1. Cylinder loaded by axial force.

20

V. Birman / International Journal of Engineering Science 78 (2014) 18–26

corresponding to buckling of struts of the foam and subsequent densification cannot be allowed. Accordingly, the application of foam would be limited to strains corresponding to the linear elastic response with the requirement that the foam should not collapse, while absorbing the maximum amount of energy during deformation. A typical goal of a static or dynamic analysis of a functionally graded cylinder would be finding an optimum density distribution to minimize the stresses or maximize the absorbed energy, while maintaining a desirable weight. Accordingly, the following sections demonstrate the solution for stresses and displacements for the static problem as well as the energy absorption in the case where the cylinder is constrained against radial displacements at the inner and outer surfaces. In the dynamic case, we demonstrate the approach to the solution of the problem of response of the cylinder subjected to a periodic-in-time axial load. The treatment of the energy dissipation problem in the dynamic case would require us to combine the solution for the dynamic response with presently unavailable analytical or empirical relations for the loss factor of the material as a function of the local mass density. Alternatively, if the loss factor of the cylinder evaluated on the macroscopic scale was available, it could be used in an analytical solution.

3. Static problem: functionally graded cylinder subject to an axial force – stress analysis and energy absorption In the axisymmetric formulation, the equation of equilibrium in the axial direction is

@ rr rr  rh þ ¼0 @r r

ð1Þ

where rr and rh are radial and circumferential stresses, respectively. The strain–displacement relationships are

er ¼

u r

@u ; @r

eh ¼ ; ez ¼

@w @z

ð2Þ

In Eq. (2), er, eh and ez are radial, circumferential and axial strains, respectively, u is the radial displacement and w is the axial displacement. According to the first assumption, the axial displacement is independent of the radial coordinate, i.e. w = w(z). Constitutive relations for the isotropic material are:

8 9 > < rr > = > :

2

C 11

C 12

rh ¼ 6 C 11 4 > rz ; sym

38 9 C 12 > < er > = 7 C 12 5 eh > : > C 11 ez ;

ð3Þ

where Cij are the elements of the tensor of stiffness. Shear stresses and strains are absent in the axisymmetric plane-strain problem without warping considered here. The stiffness coefficients for an isotropic material are expressed in terms of Lame’s constants as follows:

C 11 ¼ k þ 2G;

C 12 ¼ k;

k¼

Em ð1 þ mÞð1  2mÞ

ð4Þ

where E, G and m are the moduli of elasticity and shear and the Poisson ratio, respectively. As a result of material grading in the radial direction, the stiffness coefficients vary with the radial coordinate, so that in the general case the exact integration of Eq. (1) is impossible. The exact solution of a relevant problem for a cylinder with radially varying properties was obtained by Lekhnitskii (1963) for the case where m

C 11 ¼ kr ;

C 12 ¼ nr m

ð5Þ

where m is a real number. The units of the stiffness coefficients k and n should result in the appropriate units of the elements of the tensor of stiffness. An example where the grading described by Eq. (5) is plausible is found in open-cell foam. For such foam, the moduli of elasticity and shear, and the Poisson ratio can be presented by the following phenomenological relations (Gibson & Ashby, 1997):

 2 E ¼ Es

q qs

;

G¼

 2 3 q Es ; qs 8

m¼

1 3

ð6Þ

where Es and qs are the modulus of elasticity and the mass density of the solid foam material. In some closed-cell foams the modulus of elasticity was also found to be a power function of the mass density with the power equal to 1.7 (Goods, Neuschwanger, Whinnery, & Nix, 1999). Consider the case where the foam density varies with the radial coordinate according to

q ¼ k1=2 rm=2

ð7Þ

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V. Birman / International Journal of Engineering Science 78 (2014) 18–26

where k is a coefficient (the dimensions of this coefficient should be such that the units of the expression in the right side of  Eq. (7) comply with the units of mass density). Then the elastic and shear moduli are

 m; E ¼ kr

G¼

3 m kr 8

ð8Þ

where

k Es ¼ k 2

ð9Þ

qs

Eq. (8) lead to the elements of the tensor of stiffness according to Eq. (5). If open-cell foam is employed in multiple-use energy absorption applications, i.e. it is subject to a repeated load, the collapse associated with buckling of the truss should be prevented. Then the stress applied to open-cell foam should not exceed the elastic collapse stress (Gibson & Ashby, 1997):

rcollapse ¼ 0:05Es z

 2

q qs

ð10Þ

The substitution of Eqs. (2), (3), and (5) into (1) yields the equation of equilibrium in terms of the radial displacement and the axial strain: 2

d u dr

2

þ

1 þ m du 1  mn=k mn  u¼ ez r dr r2 kr

ð11Þ

Eq. (11) is a heterogeneous Bessel equation that can be solved in terms of ez:

u ¼ C 1 r s1 þ C 2 r s2 

n ez r kþn

ð12Þ

where

s1 ;2 ¼

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 mn m  m2 þ 4 1  2 k

ð13Þ

If the density decreases with radius, as is anticipated in energy absorption applications, i.e. m < 0, the roots si (i = 1, 2) are real numbers. Constants of integration C1 and C2 can be determined using the boundary conditions along the inner and outer radii of the cylinder. For example, if the cylinder is enclosed between rigid walls or layers, radial displacements are

uðr ¼ aÞ ¼ uðr ¼ bÞ ¼ 0

ð14Þ

Then, as follows from Eq. (12),

C 1 ¼ C 1 ez ;

C 2 ¼ C 2 ez



where

C1 ¼ 

C2 ¼









n kþn n kþn



ð15Þ



abs2 as2 b as1 bs2 as2 bs1 as1 babs1 s1 s2 s2 s1

 

ð16Þ

a b a b

Accordingly, the radial and tangential strains can be expressed in terms of the axial strain:





n er ¼ C 1 s1 rs1 1 þ C 2 s2 rs2 1  kþn ez





ð17Þ

n eh ¼ C 1 rs1 1 þ C 2 rs2 1  kþn ez

The axial force in the cylinder is related to the axial stresses by

P ¼ 2p

Z

b

rz rdr

ð18Þ

a

The longitudinal stress is given by the last Eq. (3) which upon using Eq. (5) and the substitution of radial and tangential strains in terms of the axial strain per Eq. (17) becomes



rz ¼ nC 1 ðs1 þ 1Þrs1 þm1 þ nC 2 ðs2 þ 1Þrs2 þm1 þ

kðk þ nÞ  2n2 m r kþn



ez

ð19Þ

The integration of Eq. (18) yields the explicit relationship between the applied force and axial strain:

"

s þmþ1

P ¼ 2p nC 1 ðs1 þ 1Þ 

b1

s þmþ1

mþ2

 as1 þmþ1 b2  as2 þmþ1 kðk þ nÞ  2n2 b  amþ2 þ nC 2 ðs2 þ 1Þ þ  s1 þ m þ 1 s2 þ m þ 1 kþn mþ2

#

ez

ð20Þ

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V. Birman / International Journal of Engineering Science 78 (2014) 18–26

Now it is possible to express all strains and stresses in terms of the applied force since the axial strain ez is available as a function of P from Eq. (20). The amount of energy absorbed by the cylinder subject to a static axial force is given by

P ¼ Pez L

ð21Þ

where the product of the axial strain ez and length L represents the axial displacement of the loaded end. The substitution of Eq. (20) into (21) yields an explicit expression for the absorbed energy

P¼

P2 L 1 2p nC 1 ðs1 þ 1Þ bs1 þmþ1 as1 þmþ1 þ nC 2 ðs2 þ 1Þ bs2 þmþ1 as2 þmþ1 þ kðkþnÞ2n2 s1 þmþ1 s2 þmþ1 kþn 



bmþ2 amþ2 mþ2

ð22Þ

While the absorption of energy can be evaluated using Eq. (22) for a prescribed power m, a desirable design can be identified only taking its weight into consideration. The latter can be found from

GðmÞ ¼ 2pLg

Z

m=2þ2

b

a

qðrÞrdr ¼ 2pk1=2 Lg

b

 am=2þ2 m=2 þ 2

ð23Þ

where g is the gravity acceleration. 4. Dynamic problem: functionally graded cylinder subject to a periodic-in-time axial force Consider forced vibrations of a cylinder with a varying in the radial direction mass density. The cylinder is subject to axial periodic-in-time force P = P0 sin xt. In the following we determine displacements during a cycle of vibration by assumption that damping does not noticeably affect the forced motion. Once displacements are determined, the strains and stresses are available. While the problem of energy dissipation in the dynamic problem cannot be addressed without data on the effect of material density on the loss factor, we note two possible strategies for a graded cylinder. If the overall effective viscous damping coefficient of the cylinder can be specified from theoretical or experimental analyses, the solution presented below can be extended to account for the dissipation during one cycle of vibrations. This approach can be characterized as macromechanical since it does not concentrate on local material contributions to damping, rather analyzing the cylinder as a whole. An alternative micromechanical approach would specify the damping coefficient as an analytical function of the local mass density of the material. Subsequently, a local dissipation can be integrated over the volume occupied by the cylinder to yield the total energy dissipation. Two possible modes of failure of a cylinder that is dynamically loaded in the axial direction include the loss of strength and dynamic instability. Various aspects of dynamic instability of composite cylindrical shells under periodic in time axial loads have been extensively investigated (e.g., Bert & Birman, 1988; Lam & Ng, 1998; Darabi et al., 2008). Note that foams often display a different response in tension and compression. Accordingly, the following dynamic analysis assuming identical tensile and compressive stress–strain behavior of the functionally graded material may have to be modified if applied to foams. Such modification would rely on the approach similar to that for a bimodulus material (e.g., Bert & Kumar, 1982). In the absence of damping the equation of motion in the axial direction is

l

@ 2 wðz; tÞ ¼ 2p @t2

Z a

b

@ rz ðr; z; tÞ rdr @z

ð24Þ

where l is the mass per unit length of the cylinder and t is time. Axisymmetric radial vibrations of the cylinder are described by the following equation:

@ rr rr  rh @2u þ ¼q 2 @r r @t

ð25Þ

The steady-state axial motion is represented by

wðz; tÞ ¼ WðzÞ sin xt

ð26Þ

where we account for the absence of warping (i.e. the axial motion depends only on the axial coordinate and time, while being independent of the radial coordinate). The radial vibration is represented by

u ¼ uðr; z; tÞ ¼ Uðr; zÞ sin xt

ð27Þ

The substitution of Eq. (27) into (25) and using Eqs. (3) and (5) yields

  @ 2 U 1 þ m @U 1  mn=k Sx2 mn @W U¼ þ   @r 2 r @r r2 kr @z r m=2

ð28Þ

V. Birman / International Journal of Engineering Science 78 (2014) 18–26

23

Let

@W ¼ f ðzÞ; @z

Uðr; zÞ ¼ UðrÞf ðzÞ

ð29Þ

Then Eq. (28) becomes

  1 þ m dU 1  mn=k Sx2 mn   U¼ r dr r2 kr r m=2

2

d U dr

2

þ

ð30Þ

The solution of the homogeneous version of Eq. (30) is found in the form (Bowman, 1958):

U hom ðrÞ ¼ ra ½H1 J n ðbrc Þ þ H2 F n ðbr c Þ ¼ H1 u1 ðrÞ þ H2 u2 ðrÞ

ð31Þ





where H1 and H2 are constants of integration, J n is the Bessel function of the first kind and the function F n depends on n being   integer of noninteger, i.e.

F n ðbr c Þ ¼ Y n ðbr c Þ 

ð32Þ



in the former case, Y n being the Bessel function of the second kind, and 

F n ðbr c Þ ¼ J  n ðbr c Þ 

ð33Þ



in the latter case. The coefficients in Eq. (31)–(33) are related to the coefficients in Eq. (30) by

m 2

m þ 1; 4

a¼ ; c¼

b2 ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 2

x 1 mn m2 1 S; n ¼ þ  c k c 4

ð34Þ

Note that if the mass density of the foam decreases toward the outer radius, i.e. m < 0, n is always real.  The solution of the nonhomogeneous Eq. (30) can be obtained in terms of the particular solution by the method of variation of parameters (Lagrange’s method). Accordingly, the general solution is

UðrÞ ¼ H1 u1 ðrÞ þ H2 u2 ðrÞ  u2 ðrÞ

Z

u1 ðrÞmn dr þ u1 ðrÞ krW u1 u2

Z

u2 ðrÞmn dr krW u1 u2

ð35Þ

where the Wronskian

W u1 u2 ¼ u1 ðrÞ

@u2 ðrÞ @u1 ðrÞ  u2 ðrÞ @r @r

ð36Þ

The axial motion equation (24) yields

" k

b

# 2  amþ1 d W þ nðF 1 þ F 2 Þ þ l x2 W ¼ 0 2 mþ1 dz

mþ1

ð37Þ

where

F1 ¼

Z

b

rm a

dU dr; dr

F2 ¼

Z

b

r m1 Udr

ð38Þ

a

The solution of Eq. (37) is

W ¼ AeKz þ BeKz

ð39Þ

where A and B are constants of integration and

K¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lx2 kb

mþ1

amþ1 mþ1

ð40Þ

þ nðF 1 þ F 2 Þ

The solution given by Eq. (39) can be represented in trigonometric or hyperbolic functions dependent on K being a real or imaginary number. The boundary conditions at the ends of the cylinder yield the constants of integration in Eq. (39):

wðz ¼ 0Þ ¼ 0

A ¼ B

!

ð41Þ

and

Pðz ¼ L; tÞ ¼ 2p

Z

b

rz ðr; z ¼ L; tÞrdr a

!

P0 i A ¼ h mþ1 mþ1 b a K k mþ1 þ nðF 1 þ F 2 Þ coshðKLÞ

ð42Þ

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V. Birman / International Journal of Engineering Science 78 (2014) 18–26

Both displacements as well as the stresses are determined throughout the cylinder using this solution. If the strength is available as a function of local mass density, the optimization of the cylinder based on the stress distribution is also possible. 5. Example: absorption of energy in a functionally graded open-cell foam cylinder under axial static loading In this example we analyze the effect of foam grading in the radial direction on the energy absorption in a cylinder subjected to axial static loading. Besides the energy absorption effectiveness, we are concerned with the effect of grading on the weight of the cylinder. The problem being linear, we are able to present the considered effects in a nondimensional form, so that the amplitude of the load, the length of the cylinder and the properties of the solid foam material do not have to be specified. As a result, the conclusions obtained in the paper remain valid in a general case. Consider the situation where the mass density of the foam at the inner and outer radii of the cylinder is presented as a fraction of the mass density of the solid material, i.e.

qðaÞ ¼ i1 qs ; qðbÞ ¼ i2 qs

ð43Þ

In theory, the numbers i1 and i2 can vary in the range (0.0, 1.0), but practically they are limited within a more narrow range since the foam of a low density (q ? 0) can succumb to excessive stresses or cause warping of the cross section, while combining graded material with a layer of solid foam material (q = qs) is not considered in this study. The relative energy absorption defined as a ratio of the absorbed energy in a functionally graded cylinder to the energy absorbed by the homogeneous cylinder of the same geometry with a mass density q = i3qs(m = 0) is obtained by

PðmÞ

RðmÞ ¼

Pðm ¼ 0Þ "

2

¼

ðb  a2 Þ½kð0Þ2 þ kð0Þnð0Þ  nð0Þ2  kð0Þ þ nð0Þ s ðmÞþmþ1

s ðmÞþmþ1

 as1 ðmÞþmþ1 b2  as2 ðmÞþmþ1 þ nðmÞC 2 ðmÞ½s2 ðmÞ þ 1   s1 ðmÞ þ m þ 1 s2 ðmÞ þ m þ 1 #1 2 mþ2 kðmÞ½kðmÞ þ nðmÞ  2nðmÞ b  amþ2 þ kðmÞ þ nðmÞ mþ2  nðmÞC 1 ðmÞ½s1 ðmÞ þ 1

b1

ð44Þ

where k(0) = k(m = 0), n(0) = n(m = 0). Similarly to numbers i1 and i2, the number i3 can vary in the range from zero to unity (solid material). Given the cylinder radii and the mass density at the boundaries, i.e. a, b, i1, i2, one can determine m; kðmÞ. The density Eq.  (7) at the inner and outer radii of the cylinder in conjunction with Eq. (43) yield

m¼2

logði1 =i2 Þ ; logða=bÞ

2

kðmÞ ¼ i2 q2s b

m

ð45Þ



Now k(m) can be found from Eq. (9) and R(m) from Eq. (44). The ratio of the weight of the graded cylinder whose mass density varies according to Eqs. (7) and (43) to that of the cylinder with a constant mass density is obtained from Eqs. (23) and (45) as m=2þ2

QðmÞ ¼

m=2þ2

m=2

GðmÞ b  am=2þ2 2b i2 2 1  ða=bÞ ¼ ¼ 2 2 Gðm ¼ 0Þ m=2 þ 2 b  a2 i3 m=2 þ 2 1  ða=bÞ

i2 i3

ð46Þ

Of course, a desirable outcome of functional grading can be achieved if R(m) > 1, Q(m) 6 1. The effects of the ratio of inner-to-outer radii and functional grading on the absorbed energy and weight of the cylinder are demonstrated in Fig. 2 where i1 = 0.7, i2 = 0.5 and i3 = 0.6. As is observed in this figure, both the energy absorption and weight savings improve in a ‘‘thicker’’ shell, i.e. at smaller ratios a/b. A sharper grading of the material corresponding to i1 = 0.8, i2 = 0.4 further improves the energy absorption and marginally decreases the weight of the cylinder (Fig. 3). More

1.3 1.2

R

1.1

R,Q

1

Q

0.9 0.8 0.1

0.2

0.3

0.4

a/b

0.5

0.6

0.7

0.8

0.9

Fig. 2. Energy absorption and weight reduction in a functionally graded cylinder (i1 = 0.7, i2 = 0.5) normalized by the energy absorption and weight reduction in a homogeneous cylinder (i3 = 0.6) as functions of the inner-to-outer radii ratio.

V. Birman / International Journal of Engineering Science 78 (2014) 18–26

25

2 1.5

R

R,Q 1

Q

0.5 0.1

0.2

0.3

0.4

0.5

a/b

0.6

0.7

0.8

0.9

Fig. 3. Energy absorption and weight reduction in a functionally graded cylinder (i1 = 0.8, i2 = 0.4) normalized by the energy absorption and weight reduction in a homogeneous cylinder (i3 = 0.6) as functions of the inner-to-outer radii ratio.

2 1.8 1.6

R

R,Q 1.4 1.2 1

Q

0.8 0.5

0.55

0.6

0.65

i1

0.7

0.75

0.8

0.85

0.9

Fig. 4. Effect of the material density at the inner radius on the normalized energy absorption and weight for a/b = 0.5. The mass density at the outer radius is constant i2 = 0.4. The energy absorption and weight are normalized by those in a homogeneous cylinder with i3 = 0.6.

2

R

1.5

R,Q Q

1 0.5 0.5

0.55

0.6

0.65

i1

0.7

0.75

0.8

0.85

0.9

Fig. 5. Effect of the material density at the inner radius on the normalized energy absorption and weight for a/b = 0.2. The mass density at the outer radius is constant i2 = 0.4. The energy absorption and weight are normalized by those in a homogeneous cylinder with i3 = 0.6.

extreme grading cases have not been considered since they would likely result in warping of the cross sections perpendicular to the cylinder axis violating the assumption adopted for the present analysis. Furthermore, very low local material density in a part of the cross section would cause the loss of strength at even moderate loads. The effect of the material density at the inner radius on the energy absorption is further elucidated in Fig. 4 for the inner-to-outer radius equal to 0.5. While, the mass density at the inner radius varies, the density at the outer radius is kept constant, i2 = 0.4. A higher density in the vicinity to the inner radius of the shell is detrimental both reducing the energy absorption and increasing weight. This could be expected since a stiffer inner section of the cylinder corresponds to the overall higher stiffness. The change of the inner-to-outer radius to 0.2 does not alter the conclusion, as is demonstrated in Fig. 5. Note that it would be mistaken to outright conclude that the material density should be reduced throughout the entire cylinder to achieve the maximum energy absorption. The limiting factors are the stress and strength distributions throughout the cylinder. Accordingly, a comprehensive analysis should combine the present solution with the investigation of the local stresses affected by the grading as well as by the strength distribution. The latter requires us to specify the strength as a function of the mass density of the material. 6. Conclusions Exact solutions to the problems of stress and displacement distribution and energy absorption in a functionally graded cylinder subject to a static axial force are obtained in the case where the stiffness varies as a power function of the radial coordinate. It is shown that such grading is achievable if the cylinder is manufactured from open-cell foam. The dynamic response of the cylinder is also considered in the case where it is loaded by a periodic-in-time force. Numerical results are demonstrated for the energy absorption in the static case. It is shown that a relatively higher energy absorption and

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