Linear Momentum Principle Equation of Motion Momentum Principle

22 The momentum principle states that the time rate of change of the total momentum of a given set of particles equals the vector sum of all external forceps acting on the particles of the set, provided Newton's Third Law applies. The continuum form of this principle is a basic postulate of continuum mechanics.

Then we substitute ti = Tijnj and apply the divergence theorm to obtain dTi i v \ dxj

or for an arbitrary volume dTij dvi __ , dv

which is Cauchy's (first) equation of motion, or the linear momentum principle, or more simply equilibrium equation.

23 When expanded in 3D, this equation yields:

dTii

dTi2

dTi3

+

+

dxi

dx2

dx3

dT2i

dT22

dT23

dxi

+

dx2

+

dx3

dT3i

+

dT32

+

dT33

dxi

dx2

24 We note that these equations could also have been derived from the free body diagram shown in Fig. 6.2 with the assumption of equilibrium (via Newton's second law) considering an infinitesimal element of dimensions dxi x dx2 x dx3. Writing the summation of forces, will yield

where p is the density, bi is the body force (including inertia).

Example 6-1: Equilibrium Equation

In the absence of body forces, does the following stress distribution x\ + v (xi— xX)

-2vxix2

where v is a constant, satisfy equilibrium? Solution:

j dxj dT

dx2 dT-2 dx-dT32 dx2

dx3 dT23 dx3

dT33 dx3

Therefore, equilibrium is satisfied.

8c yy yy

8t yx

8 cx

Figure 6.2: Equilibrium of Stresses, Cartesian Coordinates 6.3.2 Moment of Momentum Principle

25 The moment of momentum principle states that the time rate of change of the total moment of momentum of a given set of particles equals the vector sum of the moments of all external forces acting on the particles of the set.

26 Thus, in the absence of distributed couples (this theory of Cosserat will not be covered in this course) we postulate the same principle for a continuum as

6.3.2.1 Symmetry of the Stress Tensor

27 We observe that the preceding equation does not furnish any new differential equation of motion. If we substitute tn = Tn and the symmetry of the tensor is assumed, then the linear momentum principle (Eq. 6.24) is satisfied.

28 Alternatively, we may start by using Eq. 1.18 (ci = £ijkajbk) to express the cross product in indicial form and substitute above:

/(8rmnXmtn)dS + I (8rmnXmbnp^)dV J( I (8rmnXmpvn)dV S J V dtJ V

we then substitute tn = Tjnnj, and apply Gauss theorem to obtain

dXmT jn j

+ xmpbn

BXn dV

8rmn~T¡ (xmvn) pdV V dt but since df^x ^^ / dt = vm, this becomes

m but £rmnvmvn = 0 since vmvn is symmetric in the indeces mn while ermn is antisymmetric, and the last term on the right cancels with the first term on the left, and finally with 5mjTjn = Tmn we are left with

^rmnTmndV 0

y or for an arbitrary volume V, at each point, and this yields

for r =

T23

— T32

= 0

for r =

1 2

T31

Ti3

= 0

for r =

3

T12

— T21

establishing the symmetry of the stress matrix without any assumption of equilibrium or of uniformity of stress distribution as was done in Sect. 2.3.

29 The symmetry of the stress matrix is Cauchy's second law of motion (1827).

0 0

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