Generalized Hookes Law

L. Narasimhulu TammineniJavier S. Daz

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DefinitionIn the 1-D case, for a linear elastic material the stress is proportional to the strain , that is =E, where the proportionality factor E is called modulus of elasticity, which is a property of the material. Thus, for 1-D case, only one material property is required to relate stress and strain for linear elastic behavior.The relation =E is known as Hookes law.Since we consider that the continuum material is a linear elastic material, we introduce the generalized Hookes Law in Cartesian coordinates

klijklij lc= (1)

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Definition (Cont)The Hookes law is a statement that the stress is proportional to the gradient of the deformation occurring in the material.These equations assume that a linear relationship exists between the components of stress tensor and strain tensor.Such relations are referred to as a set of constitutive equations. They relate stress and strain, because they depend on the material behavior, whether it be an elastic or plastic solid or a viscous fluid.In this presentation we will only consider the constitutive equations for an elastic solid.

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Constitutive EquationsThey are applicable for materials exhibiting small deformations when subjected to external forces.The 81 constants Cijkl are called the elastic stiffness of the material and are the components of a Cartesian tensor of the fourth order.It is the elastic stiffness tensor which characterizes the mechanical properties of a particular anisotropic Hookean elastic solid. The anisotropy of the material is represented by the fact that the components of Cijkl are in general different for different choices of coordinate axes. If the body is homogeneous, that is, the mechanical properties are the same for every particle of the body, then Cijkl are constants (i.e. independent of position).We shall only study homogeneous bodies.Due to the symmetry of the stress and strain tensor we find that elastic stiffness tensor must satisfy the relation

Cijkl= Cjikl= Cijlk= Cjilkand consequently only 36 of the 81 constants are actually independent.

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Material SymmetriesThe generalized Hookes law can be expressed in a form where the 36 independent constants can be examined in more detail under special material symmetries.

(2)

=

6

5

4

3

2

1

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

6

5

4

3

2

1

llllll

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

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Material SymmetriesThe 14 Bravais space latticesP = Primitive or simpleI = Body-centered cubicF = Face-center cubicC = Base-centered cubic

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Material SymmetriesFor example, for the anisotropic case, starting with 36 constants Cij, but of these are 6 where i=j. This leaves 30 where ij, but only half of these are independent, since Cij= Cji. Therefore, for the general anisotropic linear elastic solid there are 30/2+6=21 independent elastic constants.As a result of symmetry conditions found in different crystal structures the number of independent elastic constants can be reduced still further

2Isotropic

34 threefold rotationsCubic

51 sixfold rotationHexagonal

61 fourfold rotationTetragonal

92 perpendicular twofold rotationsOrthorhombic

131 twofold rotationMonoclinic

21NoneTriclinic

Number of ConstantsRotational SymmetryCrystal Structure

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Anisotropic Material

Generally, in the 3-D case Hookes law asserts that each of the stress components is a linear function of the components of the strain tensor, where the 36 independent constants, C11,,C66, are called elastic coefficients. Materials that exhibit such stress-strain relations involving a number of coefficients are said to be anisotropic. In reality, this is an assumption that is reasonably accurate for many materials subjected to small strains. For a given temperature, time and location in the body, the coefficients Cij are constants that are characteristics of the material.From an examination of the equations, we find that for an anisotropic material (crystal), with one plane of symmetry, the 36 independent constants Cij reduce to 21 and the generalized Hookes law (constitutive equation) has the form of equation (3)

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Anisotropic Material (Cont)

(3)

=

6

5

4

3

2

1

66

5655

464544

36353433

2625242322

161514131211

6

5

4

3

2

1

llllll

CCCsymmCCCCCCCCCCCCCCCCCC

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Anisotropic Material (Cont)Elastic deformation under anisotropic conditions is described by the elastic constants Cij, whose number can vary from 21 for the most anisotropic solid to 3 for one exhibiting cubic symmetry. As we are going to see next, for isotropic solids, the number of independent elastic constants is 2.There are two sources of anisotropy: Texture, in which the grains are not randomly oriented, but have one or

more preferred orientations. Texturing is often introduced by deformation processes, such as cold rolling, wire drawing, and extrusion.

Alignment of inclusions or second-phase particles along specific directions. When steel is produced, the inclusions existing in the ingot take the shape and orientation of the rolling. These inclusions produce mechanical effects called fibering.

Anisotropy can strongly affect the yield stress and also influence fracture. Some anisotropic materials, such as wood and fiber-reinforced composites, may have low strength in the radial direction.

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Orthotropic MaterialMaterials such as wood, laminated plastics, cold rolled steels, reinforced concrete, various composite materials such as laminated composites made by the consolidation of prepegged sheets, with individual plies having different fiber orientation, and even forgings can be treated as orthotropic.They possess 3 orthogonal planes of material symmetry and three corresponding orthogonal axes called orthotropic axes. In some materials (forgings) these axes may vary from point to point.In other materials (fiber-reinforced plastics, reinforced concrete), orthotropic directions remain constant as long as the fibers and steel reinforcing bars maintain constant directions. In any case, for an elastic orthotropic material, independent constants Cij remain unchanged at a point under a rotation of 180 about any of the orthotropic axes. Then, the original 36 constants Cij reduce to 12 and the generalized Hookes law (constitutive equation) has the form of equation (4)

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Orthotropic Material (Cont)

(4)

=

6

5

4

3

2

1

66

55

44

333231

232221

131211

6

5

4

3

2

1

000000000000000000000000

llllll

CC

CCCCCCCCCC

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Orthotropic Material / ExampleA lamina (a thin plate, sheet, or layer of material) of a section of an airplane wing is composed of unidirectional fibers and a resin matrix that bonds the fibers. Let the volume fraction (the proportion of fiber volume to the total volume of the composite) be f. Determine the effective linear stress-strain relations of the lamina.

Figure 1 Lamina: fiber volume fraction = f , resin volume fraction = 1 - f

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Orthotropic Material / ExampleLet the modulus of elasticity and the Poisson ratio of the fibers be denoted EF and F respectively, and the modulus elasticity and the Poisson ratio of the resin be ER and R. Since the lamina is thin, the effective state of stress in the lamina is approximately one of plane stress in the x-y plane of the lamina (see Figure 1a). Hence, the stress-strain relations for the fibers and the resin are

(a)

)(1

)(1

)(1

)(1

xxRRyyRR

yyR

yyRRxxRR

xxR

xxFFyyFF

yyF

yyFFxxFF

xxF

El

El

El

El

=

=

=

=

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Orthotropic Material / Examplewhere( xxF,yyF) ,(xxR ,yyR ),(xxF,yyF ), and (xxR,yyR) denote stress and strain components in the fiber (F) and resin (R), respectively.Since the fiber and resin are bounded, the effective lamina strain xx is the same as that in the fibers and in the resin; that is ,in the x direction

(b)In the y direction, the effective lamina strain yy is proportional to the

amount of fibre per unit length in the y direction and the amount of resin per unit length in the y direction .Hence,

(c)

Also, by equilibrium of the lamina in the x direction, the effective lamina stress xx is

(d)

xxRxxFxx ==

yyRyyFyy ff )1( +=

xxRxxFxx ff )1( +=

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Orthotropic Material / Examplein the y direction, the effective lamina stress yy is the same as in the fibers and in the resins; that is,

(d)Solving Eqs (a) through (e) for xx and yy in terms of xx and yy, we obtain the effective stress-strain relations for the lamina as

(f)

where

(g)

yyRyyFyy ==

)(1

)(1

xxyyyy

yyxxxx

E

E

=

=

]1

12)1()1)[(1(

)1()1(

22

ff

ff

EE

EEff

ffEffEE

RFF

RF

R

FR

RF

RF

++++=

+=+=

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Orthotropic Material / ExampleTo determine the shear stress-strain ,we apply a shear stress xy to a rectangular element of the lamina (Figure 1b), and we calculate the angle change xy of the rectangle. By figure 1b, the relative displacement b of the top of the element is

(h)where F and R are the angle changes attributed to the fiber and the resin, respectively; that is,

(i)

and GF and GR are the shear moduli of elasticity of the fiber and resin, respectively. Hence, the change in the angle of the element (the shear strain) is, with eqs (h) and (i),

(j)

RF ffb )1( +=

R

xyR

F

xyF GG

== ,

xyRF

FRxyxy GG

GffGb ])1([1

2 +===

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Orthotropic Material / ExampleBy Eq (j), the shear stress strain relation is

(k)where

(l)

Thus, by Eqs (f), (g), (k), (l), we obtain the stress-strain relations of the lamina, in the form of Eqs.

(m)

FR

RF

GffGGGG

)1( +=

xyxyxy GG 2==

xyxy

yyxxyy

yyxxxx

CCCCC

33

2212

1211

=+=+=

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Orthotropic Material / ExampleWhere

(n)

Finally, in an orthotropic material, if the constants Cij are written in the form of elastic moduli (E) and Poisson ratios (), it is possible for some of the Poisson ratios to exceed 0.5, which is not possible for an isotropic material.

GCEC

ECEC

====

33222

212211

,

,

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Isotropic MaterialIf the constituents of the material of a solid member are distributed sufficiently randomly, any part of a the member will display essentially the same material properties in all directions. If a solid member is composed of such randomly oriented constituents, it is said to be isotropic. Accordingly, if a material is isotropic, its physical properties at a point are invariant under a rotation of axes and is said to be elastically isotropic if its characteristic elastic coefficients Cij are invariant under any rotation of coordinates.Since isotropic materials are elastically the same in all directions and there is no directional variation on property, we can obtain

C11 = C22 = C33 = C1 (5)C12 = C13 = C23 = C2 (6)

This reduces the number of constants to 2

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Isotropic Material (Cont)In the principal coordinates system we can then write stresses as

(7)

(8)

(9)These equations can be written in terms of the index notation as

(10)

Where and(11)

are called Lames constants

)(

)(

)(

'22

'112

'331

'33

'11

'332

'221

'22

'33

'222

'111

'11

llClCllClCllClC

++=++=++=

'''' 2 ijijiiij ll +=2C=

221 CC =

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Isotropic Material (Cont)Transforming the primed quantities to unprimed ones through the use of Cartesian tensor transformation laws, we obtain

(12)

Solving for lij, we have

(13)

Equations (12) and (13) are two forms of the generalized Hookes law for an isotropic elastic solid in terms of Lamesconstants and . However, is not easy to interpret physically, and engineers frequently prefer to have the same equations expressed in terms of the engineering constants and E.

mnmniimn ll 2+=

ijijmmijl

21

)23(2++=

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Isotropic Material (Cont)Poissons ratio ()Is the ratio of the lateral strain to the longitudinal strain in a uniaxial state of stress.Youngs modulus (E)Is also called the modulus of elasticity, and is the ratio of stress to strain in a uniaxial state of stress.The strain-stress relations for and isotropic, elastic solid in terms of and E can be derived

(14)

(15)

(16))]([1

)]([1

)]([1

22113333

11332222

33221111

+=

+=

+=

El

El

El

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Isotropic Material (Cont)

Comparing the previous equations, we note that(17)

(18)

(19)

(20))1(2

)21)(1(

)(2

)23(

+=+=

+=++=

E

E

E

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Isotropic Material (Cont)

And that

(21)

(22)

(23)1212

12

3131

31

2323

23

12

12

12

El

El

El

+==

+==

+==

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Isotropic Material (Cont)Expressed in indicial form, equations (21), (22) and (23) took the form

(24)

Finally, expressing ij in terms of lij and the elastic constants and E, we find

(25)

We now have four standard forms for the generalized Hookes law for an isotropic, linear, elastic solid, equations (12), (13), (24) and (25).Note that there are only two independent constants (either and , or and E) relating the stress tensor and the strain tensor for an isotropic, homogeneous, linear elastic solid.

kkijijij EEl += )1(

]21

[1 ijkkijij

llE

+=

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Isotropic Material (Cont)The tensor equation (25) represents the six scalar equations

(26)

(27)

(28)

(29) (30) (31)

)]()1[()21)(1(

)]()1[()21)(1(

)]()1[()21)(1(

11223333

11332222

33221111

lllE

lllE

lllE

+++=

+++=

+++=

1212 1lE += 1313 1 l

E += 2323 1 l

E +=

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Isotropic Material (Cont)In matrix form

(31)

+=

31

23

12

33

22

11

31

23

12

33

22

11

22100000

02210000

00221000

000100010001

)21)(1(

llllll

E

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Homework Problem

An hydrostatic compressive stress applied to a material with cubic symmetry results in a dilatation of 10-5. The three independent elastic constants of the material are

C11=50 GPaC12=40 GPaC13=32 GPa

a) Write an expression for the generalized Hookes Lawb) Compute the applied hydrostatic stress.

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ReferencesBatra, Romesh C., Introduction to Continuum Mechanics, Virginia Polytechich Institute and University, 2000Boresi, Arthur P. and Schmidt Richard J., Advanced Mechanical of Materials, 6th Edition, John Wiley & Sons, Inc., 2003Cook, Robert D. and Young, Warren C., Advanced Mechanics of Materials, 2nd Edition, Prentice Hall, 1999Dieter, George E., Mechanical Metallurgy, 3rd Edition, McGraw-Hill, 1986 Frederick, Daniel and Chang, Tien Sun, Continuum Mechanics, Scientific Publishers, Inc. Cambridge, 1972Heinbockel, J.H., Introduction to Tensor Calculus and Continuum Mechanics, Department of Mathematics and Statistics, Old Dominion University, 1996Meyers, Marc A. and Chawla, Krishan K., Mechanical Behavior of Materials, Prentice Hall, 1999.Saouma, Victor E., Introduction to Continuum Mechanics and Elements of Elasticity/Structural Mechanics, Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, 1998