1855_PDF_C03.PDF

Stress Princ iples

3.1

Body and Surface Forces, Mass Density

Stress is a measure of

either within or on the bounding surface

of a body subjected to loads. It should be noted that in continuum mechanics

a body is considered stress free if the only forces present are those inter-

atomic forces required to hold the body together. And so it follows that the

stresses that concern us here are those which result from the application of

forces by an external agent.

Two basic types of forces are easily distinguished from one another and

are deﬁned as follows. First, those forces acting on all volume elements, and

distributed throughout the body, are known as

force intensity,

. Gravity and inertia

forces are the best-known examples. We designate body forces by the vector

symbol

body forces

(force per unit volume).

Second, those forces which act upon and are distributed in some fashion

over a surface element of the body, regardless of whether that element is

part of the bounding surface, or an arbitrary element of surface within the

body, are called

(force per unit mass), or by the symbol

, and

have dimensions of force per unit area. Forces which occur on the outer

surfaces of two bodies pressed against one another (contact forces), or those

which result from the transmission of forces across an internal surface are

examples of surface forces.

Next, let us consider a material body

surface forces.

These are denoted by the vector symbol

having a volume

enclosed by a

surface

, and occupying a regular region

of physical space. Let

be an

interior point of the body located in the small element of volume

∆

whose

mass is

as indicated in Figure 3.1 . Recall that mass is that property of a

material body by virtue of which the body possesses inertia, that is, the

opposition which the body offers to any change in its motion. We deﬁne the

average density

∆

of this volume element by the ratio

= ∆

∆

ρ ave

(3.1-1)

FIGURE 3.1

Typical continuum volume

with element

∆

having mass

∆

at point

. Point

would be

in the center of the inﬁnitesimal volume.

and the density

at point

by the limit of this ratio as the volume shrinks

to the point

∆

ρ =

lim

∆

(3.1-2)

→

The units of density are kilograms per cubic meter (kg/m

). Notice that the

two measures of body forces,

having units of Newtons per kilogram

(N/kg), and

having units of Newtons per meter cubed (N/m

), are related

through the density by the equation

(3.1-3)

Of course, the density is, in general, a scalar function of position and time

as indicated by

(

)or

(

)

(3.1-4)

and thus may vary from point to point within a given body.

3.2

Cauchy Stress Principle

We consider a homogeneous, isotropic material body

having a bounding

surface

, and a volume

, which is subjected to arbitrary surface forces

and body forces

. Let

be an interior point of

and imagine a plane

surface

* passing through point

(sometimes referred to as a

cutting plane

)

FIGURE 3.2A

Typical continuum volume showing cutting plane

* passing through point

FIGURE 3.2B

Force and moment acting at point

in surface element

∆

so as to partition the body into two portions, designated I and II ( Figure 3.2A ) .

Point

* of the cutting plane, which is

deﬁned by the unit normal pointing in the direction from Portion I into

Portion II as shown by the free body diagram of Portion I in Figure 3.2B .

The internal forces being transmitted across the cutting plane due to the

action of Portion II upon Portion I will give rise to a force distribution on

is in the small element of area

∆

equivalent to a resultant force

∆

and a resultant moment

∆

, as is

also shown in Figure 3.2B . (For simplicity body forces

and surface forces

acting on the body as a whole are not drawn in Figure 3.2 .) Notice that

∆

and

∆

are not necessarily in the direction of the unit normal vector

. The

Cauchy stress principle

asserts that in the limit as the area

∆

* shrinks

to zero with

remaining an interior point, we obtain

∆

()

t i n

li *

(3.2-1)

∆

→

FIGURE 3.3

(

Traction vector

acting at point

of plane element

∆

, whose normal is

and

∆

li *

(3.2-2)

∆

→

()

The vector

* =

is called the

stress vector,

or sometimes the

traction

t i

vector.

the

moment vector vanishes, and there is no remaining concentrated moment,

In Eq 3.2-2 we have made the assumption that in the limit at

as it is called.

The appearance of in the symbol for the stress vector serves to remind

us that this is a special vector in that it is meaningful only in conjunction with

its associated normal vector

couple stress

()

t i

. Thus, for the inﬁnity of cutting planes

imaginable through point

, each identiﬁed by a speciﬁc , there is also an

inﬁnity of associated stress vectors for a given loading of the body. The

totality of pairs of the companion vectors

()

t i

()

and

, as illustrated by a

t i

at that point.

By applying Newton’s third law of action and reaction across the cutting

plane, we observe that the force exerted by Portion I upon Portion II is equal

and opposite to the force of Portion II upon Portion I. Additionally, from the

principle of linear momentum (Newton’s second law) we know that the time

rate of change of the linear momentum of

typical pair in Figure 3.3 , deﬁnes the

state of stress

portion of a continuum body

is equal to the resultant force acting upon that portion. For Portions I and

II, this principle may be expressed in integral form by the respective equa-

tions (these equations are derived in Section 5.4 from the principle of linear

momentum),

any

tdS+ bdV= d

∫

()

vdV

(3.2-3

)

tdS+ bdV= d

∫

()

vdV

(3.2-3

)

where

and

are the bounding surfaces and

and

are the volumes

of Portions I and II, respectively. Also,

are the body forces,

is the density,

and

is the velocity ﬁeld for the two portions. We note that

and

each

contain

* as part of their total areas.

The linear momentum principle may also be applied to the body

as a

whole, so that

tdS+ bdV= d

∫

()

vdV

(3.2-4)

If we add Eq 3.2-3

and Eq 3.2-3

and utilize Eq 3.2-4, noting that the normal

for Portion I is

, whereas for Portion II it is

– n

, we arrive at the

equation

[

]

∫

()

− ( )

S 0

(3.2-5)

*. This equation must

hold for arbitrary partitioning of the body (that is, for every imaginable

cutting plane through point

since both

and

contain a surface integral over

) which means that the integrand must be

identically zero. Hence,

()

− ( )

=−

(3.2-6)

indicating that if Portion II had been chosen as the free body in Figure 3.2

instead of Portion I, the resulting stress vector would have been

()

t i n

–

3.3

The Stress Tensor

As noted in Section 3.2, the Cauchy stress principle associates with each

direction

()

at point

a stress vector

t i

. In particular, if we introduce a

rectangular Cartesian reference frame at

, there is associated with each of

the area elements

(

= 1,2,3) located in the coordinate planes and having

()

e i

unit normals

i = 1,2,3), respectively, a stress vector as shown in

Figure 3.4 . In terms of their coordinate components these three stress vectors

associated with the coordinate planes are expressed by

(

t i

() ()

()

(3.3-1 a )

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: