MEEN 5330 Continuum Mechanics,

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MEEN 5330

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MEEN 5330 Continuum Mechanics,

all references from Continuum Mechanics by Frederick unless noted


Overview of lecture

1.
SRI survey, and quiz

2.
See web sites

3.
Start lecture on Elasticity / Solid Mechanics.


Generalized Hooke’s Law

Modified by L. Peel

L. Narasimhulu Tammineni

Javier S. Díaz

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Continuum Mechanics Web References

Best reference, Good book, but can’t print

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Continuum Mechanics Web References cont’d

Aerodynamics
-
related notes with demonstration of a “FOIL
-
sim program,

http://www1.esc.auckland.ac.nz/People/Staff/Archer/Courses/ENGSCI342.html


Undergraduate Continuum Mechanics with a Civil Engineering bent

http://www.engr.usask.ca/classes/CE/311/


Short Course Notes (gives some good examples)

http://geolab.mechan.ntua.gr/teaching/lectnotes.html#engcm


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Chapter 5, Solid Mechanics

Homework Assignment:

1.
See end of lecture


Solid Mechanics


Linear Elasticity

Study of solids following a certain elastic law, typically the material
is assumed to have a linear stress
-
strain curve in the region of
interest, or in other words, for axial tension, the
s
=E
e
, where E is
the slope of the stress
-
strain curve.

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Equation of Motion / Equilibrium

Assuming that displacements are small, where
r

= density, u
i

is
displacement, and
f
i

are the body forces, then:

Let’s look at the equations of equilibrium from Ch. 2:

Note that the only difference is the energy from movement. If we
assume that the body is moving slowly, then we can use the equations of
equilibrium.

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Equilibrium of stress and internal forces

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Small Strain


displacement & Compatibility

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Constitutive Equations (stress & strain)

For a uni
-
axial stress
-
state:

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Summary

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Anisotropic Elastic Solids

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Anisotropic Elastic Solids cont’d

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Anisotropic Elastic Solids cont’d

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Converting a stress matrix to a vector

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Converting a stress matrix to a vector cont’d

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Stress and Strain

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Special Cases

What is different here?

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Special Cases cont’d

What is different here?

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Special Cases cont’d

What is different here?

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Generalized Hooke’s Law

MEEN 5330


Modified by L. Peel

L. Narasimhulu Tammineni

Javier S. Díaz

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Definition


In the 1
-
D case, for a linear elastic material the stress
σ

is proportional to the strain
ε
, that is
σ=Eε
, where the
proportionality factor
E

is called modulus of elasticity, which
is a property of the material.

The relation
σ=Eε

is known as Hooke’s law.


Since we consider that the continuum material is a
linear elastic material, we introduce the generalized Hooke’s
Law in Cartesian coordinates

(1)

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Definition (Cont…)


Hooke’s 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 Equations


They are applicable for materials exhibiting small deformations when
subjected to external forces.



The 81 constants
C
ijkl

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
C
ijkl

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
C
ijkl

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


C
ijkl=
C
jikl=
C
ijlk=
C
jilk



and consequently only 36 of the 81 constants are actually independent.

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Material Symmetries


The generalized Hooke’s law can be expressed in a form where the 36
independent constants can be examined in more detail under special
material symmetries.




(2)

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Material Symmetries




The 14 Bravais space lattices

P = Primitive or simple

I = Body
-
centered cubic

F = Face
-
center cubic

C = Base
-
centered cubic

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Material Symmetries


For example, for the anisotropic case, starting with 36 constants
C
ij
, but
of these are 6 where
i=j
. This leaves 30 where
i
≠j
, but only half of
these are independent, since
C
ij
=
C
ji
. 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

Crystal Structure

Rotational Symmetry

Number of Constants

Triclinic

None

21

Monoclinic

1 twofold rotation

13

Orthorhombic

2 perpendicular twofold rotations

9

Tetragonal

1 fourfold rotation

6

Hexagonal

1 sixfold rotation

5

Cubic

4 threefold rotations

3

Isotropic

2

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Isotropic Material


If 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
C
ij

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








C
11

= C
22

= C
33
= C
1



(5)




C
12

= C
13

= C
23
= C
2



(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 Lame’s constants


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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
l
ij
, we have


(13)



Equations (12) and (13) are two forms of the generalized
Hooke’s law for an isotropic elastic solid in terms of Lame’s
constants
λ

and
μ
. However,
λ

is not easy to interpret physically,
and engineers frequently prefer to have the same equations
expressed in terms of the engineering constants
ν

and
E
.

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Isotropic Material (Cont…)


Poisson’s ratio (
ν
)


Is the ratio of the lateral strain to the longitudinal strain in a uniaxial state
of stress.


Young’s 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
ν

and
E
can be derived


(14)


(15)


(16)


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Isotropic Material (Cont…)

Comparing the previous equations, we note that

(17)



(18)



(19)



(20)

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Isotropic Material (Cont…)

And that


(21)



(22)



(23)



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Isotropic Material (Cont…)


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


(24)



Finally, expressing
σ
ij

in terms of
l
ij
and the elastic constants
ν

and
E
,
we find


(25)



We now have four standard forms for the generalized Hooke’s law for
an isotropic, linear, elastic solid, equations (12), (13), (24) and (25).


Note that there are only two independent constants (either
λ

and
μ
, or
ν

and
E
) relating the stress tensor and the strain tensor for an isotropic,
homogeneous, linear elastic solid.

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Isotropic Material (Cont…)


The tensor equation (25) represents the six scalar
equations


(26)


(27)


(28)







(29)




(30)


(31)

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Isotropic Material (Cont…)


In matrix form




(31)









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Anisotropic Material


Generally, in the 3
-
D case Hooke’s law asserts that each of the stress
components is a linear function of the components of the strain tensor,
where the 36 independent constants,
C
11
,…,
C
66
, 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
C
ij

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
C
ij

reduce to 21 and the generalized Hooke’s law (constitutive
equation) has the form of equation (3)



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Anisotropic Material (Cont…)






(3)

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Anisotropic Material (Cont…)


Elastic deformation under anisotropic conditions is described by the elastic
constants
C
ij
, 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 Material


Materials such as wood, laminated plastics, cold rolled steels,
reinforced concrete, various composite materials such as laminated
composites made by the consolidation of pre
-
pregged 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
C
ij

remain
unchanged at a point under a rotation of 180
°

about any of the
orthotropic axes.

Then, the original 36 constants
C
ij

reduce to 12 and
the generalized Hooke’s law (constitutive equation) has the form of
equation (4)

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Orthotropic Material (Cont…)





(4)

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Orthotropic Material / Example


A 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 / Example


Let the modulus of elasticity and the Poisson ratio of the fibers be
denoted
E
F

and
ν
F

respectively, and the modulus elasticity and the
Poisson ratio of the resin be
E
R

and
νR
. 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)



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Orthotropic Material / Example


where
(
σ
xxF

yyF
) ,(
σ
xxR


yyR

),(ε
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 ε
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 ε
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
σ
xx

is


(d)

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Orthotropic Material / Example


in the
y

direction, the effective lamina stress
σ
yy

is the same as in the
fibers and in the resins; that is,


(d)



Solving Eqs (a) through (e) for
ε
xx

and
ε
yy

in terms of
σ
xx

and
σ
yy
, we
obtain the effective stress
-
strain relations for the lamina as



(f)




where


(g)



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Orthotropic Material / Example


To determine the shear stress
-
strain ,we apply a shear stress
σ
xy
to a
rectangular element of the lamina (Figure 1b), and we calculate the
angle change γ
xy
of the rectangle. By figure 1b, the relative
displacement
b

of the top of the element is

(h)



where
γ
F

and γ
R

are the angle changes attributed to the fiber and the
resin, respectively; that is,

(i)



and G
F

and G
R

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)


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Orthotropic Material / Example


By 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)


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Orthotropic Material / Example


Where




(n)





Finally, in an orthotropic material, if the constants
C
ij

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







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Homework Problem


An hydrostatic compressive stress applied to a material with
cubic symmetry results in a cubical dilatation of

10
-
5

mm/mm.
The three independent elastic constants of the material are

C
11
=50 GPa

C
12
=40 GPa

C
66
=32 GPa


a) Write an expression for the generalized Hooke’s Law


b) Compute the applied hydrostatic stress.

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References



Batra, Romesh C., “Introduction to Continuum Mechanics”, Virginia
Polytechich Institute and University, 2000

Boresi, Arthur P. and Schmidt Richard J., “Advanced Mechanical of
Materials”, 6
th

Edition, John Wiley & Sons, Inc., 2003

Cook, Robert D. and Young, Warren C., “Advanced Mechanics of
Materials”, 2
nd

Edition, Prentice Hall, 1999

Dieter, George E., “Mechanical Metallurgy”, 3
rd

Edition, McGraw
-
Hill,
1986

Frederick, Daniel and Chang, Tien Sun, “Continuum Mechanics”,
Scientific Publishers, Inc. Cambridge, 1972

Heinbockel, J.H., “Introduction to Tensor Calculus and Continuum
Mechanics”, Department of Mathematics and Statistics, Old Dominion
University, 1996

Meyers, 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