1855_PDF_C06.PDF

Linear Elast icity

6.1

Elasticity, Hooke’s Law, Strain Energy

Elastic behavior is characterized by the following two conditions: (1) where

the stress in a material is a unique function of the strain, and (2) where the

material has the property for complete recovery to a “natural” shape upon

removal of the applied forces. If the behavior of a material is not elastic, we

say that it is

inelastic

. Also, we acknowledge that elastic behavior may be

linear

. Figure 6-1 shows geometrically these behavior patterns

by simple stress-strain curves, with the relevant loading and unloading paths

indicated. For many engineering applications, especially those involving

structural materials such as metals and concrete, the conditions for elastic

behavior are realized, and for these cases the theory of elasticity offers a very

useful and reliable model for design.

Symbolically, we write the constitutive equation for elastic behavior in its

most general form as

non-linear

()

(6.1-1)

is any one of the various

strain tensors we introduced earlier. However, for the response function

where

is a symmetric tensor-valued function and

in this text we consider only that case of Eq 6.1-1 for which the stress is a

linear

function of strain. Also, we assume that, in the deformed material, the

displacement gradients are everywhere small compared with unity. Thus, the

distinction between the Lagrangian and Eulerian descriptions is negligible, and

following the argument of Eq 4.7-3 we make use of the inﬁnitesimal strain

tensor deﬁned in Eq 4.7-5, which we repeat here:

∂

(

)

(6.1-2)

i,j

j,i

FIGURE 6.1

Uniaxial loading-unloading stress-strain curves for (a) linear elastic; (b) nonlinear elastic; and

Within the context of the above assumptions, we write the constitutive

equation for linear elastic behavior as

(6.1-3)

ijkm

where the tensor of elastic coefﬁcients

= 81 components. However,

due to the symmetry of both the stress and strain tensors, it is clear that

has 3

ijkm

(6.1-4)

ijkm

jikm

ijmk

which reduces the 81 possibilities to 36 distinct coefﬁcients at most.

by a consideration of the

elastic constitutive equation when expressed in a rotated (primed) coordinate

system in which it has the form

We may demonstrate the tensor character of

′ = ′

′

(6.1-5)

ijpn pn

But by the transformation laws for second-order tensors, along with Eq 6.1-3,

′ =

=aaC

iq js qs

iq js qskm km

aaC a a

′

iq js qskm pk nm pn

which by a direct comparison with Eq 6.1-5 provides the result

C ijpn

′ =

aaa a C

(6.1-6)

iq js pk nm qskm

that is, the transformation rule for a fourth-order Cartesian tensor.

In general, the

coefﬁcients may depend upon temperature, but here

ijkm

we assume

(constant temper-

ature) conditions. We also shall ignore strain-rate effects and consider the

components

adiabatic

(no heat gain or loss) and

isothermal

to be at most a function of position. If the elastic coefﬁcients

are constants, the material is said to be

ijkm

These constants are

those describing the elastic properties of the material. The constitutive law

given by Eq 6.1-3 is known as the

homogeneous.

generalized Hooke’s law.

For certain purposes it is convenient to write Hooke’s law using a single

subscript on the stress and strain components and double subscripts on the

elastic constants. To this end, we deﬁne

(6.1-7

)

and

= 2

(6.1-7

)

= 2

where the factor of two on the shear strain components is introduced in

keeping with Eq 4.7-14. From these deﬁnitions, Hooke’s law is now written

(6.1-8)

αβ

with Greek subscripts having a range of six. In matrix form Eq 6.1-8 appears

CCCCCC

(6.1-9)

does not constitute a tensor.

In view of our assumption to neglect thermal effects at this point, the

energy balance Eq 5.7-13 is reduced to the form

We point out that the array of the 36 constants

αβ

u= 1

ρ σ

ij ij

(6.1-10

)

which for small-deformation theory, by Eq 4.10-18, becomes

u= 1

ρ σε

(6.1-10

)

ij ij

The internal energy

in these equations is purely mechanical and is called

the

(per unit mass). Recall now that, by the continuity equation

in Lagrangian form,

strain energy

and also that to the ﬁrst order of approximation

≈+

∂

= det

det

(6.1-11)

Therefore, from our assumption of small displacement gradients, namely

∂

<< 1, we may take

≈

1 in the continuity equation to give

, a

constant in Eqs 6.1-10.

For elastic behavior under the assumptions we have imposed, the strain

energy is a function of the strain components only, and we write

(

)

(6.1-12)

so that

∂

∂ε

(6.1-13)

and by a direct comparison with Eq 6.1-10

we obtain

ρ σ

∂

∂ε

(6.1-14)

The

strain energy density,

(strain energy per unit volume) is deﬁned by

(6.1-15)

and since

, a constant, under the assumptions we have made, it follows

from Eq 6.1-14 that

σρ ∂

∂

∂ε

(6.1-16)

∂ε

It is worthwhile noting at this point that elastic behavior is sometimes

deﬁned on the basis of the existence of a strain energy function from which

the stresses may be determined by the differentiation in Eq 6.1-16. A material

deﬁned in this way is called a

material. The stress is still a unique

function of strain so that this energy approach is compatible with our earlier

deﬁnition of elastic behavior. Thus, in keeping with our basic restriction to

inﬁnitesimal deformations, we shall develop the linearized form of Eq 6.1-16.

Expanding

hyperelastic

about the origin, we have

() +

()

WW W

∂

∂ε ∂ε

() =

() +

εε

(6.1-17)

∂ε

and, from Eq 6.1-16,

== () +

() +

∂

∂ε ∂ε

∂

∂ε

(6.1-18)

∂ε

It is customary to assume that there are no residual stresses in the unstrained

state of the material so that

= 0. Thus, by retaining only the

linear term of the above expansion, we may express the linear elastic con-

stitutive equation as

= 0 when

() =

∂

∂ε ∂ε

(6.1-19)

ijkm km

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