= d dt t (t x) d t V t V (P) = d t dt +t t t v d t V t V (P) = 0 , (6-7) we
derive the local field equation as eq. (6-7) has to hold for an arbitrary
volume V(P), d t dt + t t t v = t t + t t t ( ) v = 0 . (6-8) In fluid
mechanics, the respective equation
GAUSS31, REYNOLDS32 transport equation is obtained, d dt t d t V t
V (P) = t t d t V t V (P) + t t v nd t A t A(P) . (6-5) Note, that the
domain of the integral is fixed to the body. If it is an arbitrarily moving
control volume as in fluid mechanics, a s
NEWTON. Introducing the density functions of body forces and contact
forces, eqs. (5-3) and (5-4), it follows d dt t v t d t V t V (P) = t b t d t V
t V (P) + t t n d t A t A(P) . (6-11) Introducing CAUCHYs law (5-9),
conservation of mass (6-7) and the di
volume element. Figure 5.6: Plane stress state in different coordinate
systems: (a) original (x,y), (b) principal axes, I , II , eqs. (5-38), (5-39),
(c) direction of maximum shear stress, nt = tn = max = 1 2 I ( ) II , nn =
tt = 1 2 I + ( ) II , eqs. (5-
(4-16) with t 0 F1 = 0 1 ( t x,t) t x = 0 x t x = t 0 ( ) x T = t grad 0 x (417) and 0 t F t 0 F1 = 1 (4-18) In the trivial case of 0 t F = 1 the line
element experiences no change of length or orientation, d t x = d 0 x . If
0 t F is orthogonal, i.e. 0 t
It follows from the balance of angular momentum (see section 6.3), that
t S is symmetric for non-polar media 25, t S =t ST , t ij =t ji , (5-11) and
hence t t n =t n t S = t S t n . (5-12) In a convective or material
coordinate system, eq. (4-25), the CAU
5.3 PIOLA-KIRCHHOFF Stresses Beside CAUCHY stresses, many
other stress tensors are in use. In the previous section, stress is
understood as force, d t fc , per area, d t A. of the current configuration, t
B. As the current configuration arising under the
unnecessary to account for all three components of the stress vector. In
sheet materials or under in-plane loading conditions, all stress vectors
can be assumed to lie in one plane, and the stress tensor in a Cartesian
coordinate system, ex , e y , e cfw_
, (6-15) Conservation of mass yields t d(O) (P) = t x t x t d t V t V (P) ,
(6-16) and CAUCHY's law (5-9) with divergence theorem, t x t t n d t A
t A(P) = t x t n t ( ) S d t A t A(P) = t x t ST ( ) t nd t A t A(P) = t t x t
( ) S d t V t V (P) = 2t q +t
impressis cogitur statum illum mutare. Lex II [Constat] mutationem
motus proportionalem esse vi motrici impressae, et fieri secundum
lineam rectam qua vis illa imprimitur. Lex III [Constat] actioni
contrariam semper et aequalem esse reactionem: sive corpo
CAUCHY (1789-1857) 21 For small deformations no difference has to
be made between differentiation with respect to the initial or the current
coordinates EngMech-Script.doc, 29.11.2005 - 25 - = d dt = ( 0 x,t) t 0 x
. (4-54) Since t x = t 0 ( ) x,t in the
d 0 a 0 T , (5-28) we obtain the symmetric second PIOLA-KIRCHHOFF
tensor, 0 T = J t 0 F1 t S t 0 FT = 0 TT . (5-29) Inversely, CAUCHY
stresses result from t S = 1 J 0 t F 0 T 0 t FT . (5-30) First and second
PIOLA-KIRCHHOFF tensors write as 0 t T = 0 ij 0
dilatation of a material element under small deformations, kk = trE d t V
d 0 V d 0 V , (4-52) Eq. (4-50) allows for calculating a tensor field, E,
from a given displacement field, u, uniquely. If a tensor field E is given,
it does not automatically follo
t x, eq. (4-10) determines all those particles, which pass through this
point at different instants of time, t > t0. On the other hand, at a specific
instant of time, eq. (4-10) specifies all particles that are positioned at
different points of the curren
(also addressed as test functions) meeting eq. (6-27), and is a real
number. xi = i (t) and x i = i (t) are called variations of xi and x i ,
respectively. The integral of eq. (6.26) can now be written as a function
of , I() = F t, x1 + 1,., x 1 + 1 ( ) ,
mass accelerations as negative fictitious external forces, we obtain
D'ALEMBERT's37 principle, W ex ( ) B = W in , that external and
internal forces are balanced. Special cases of the principle of virtual
work are obtained for Rigid bodies,W in = 0 , Elas
(5-15) We shall now investigate whether there exists an orientation of
the surface element at a given point, along which the stress vector is
collinear with the normal of the element, i.e. the stress vector has a
normal component only and no shear stresse
of HENCKY strains, 0 t i (H) , is that they are additive for two
subsequent deformations steps, i.e. 0 t2 i (H) = 0 t1 i (H) + t1 t2 i (H) .
GREEN's strain tensor. 0 t E(G) describes the change of the square of a
line element in the current configuration
14 CFR 43.9 and 43.11 require the technician to make appropriate
entries of maintenance actions or inspection results in the aircraft
maintenance record. 14 CFR 91.417 defines how long those records
must be kept. Whenever maintenance, preventive maintenan
t t n t v d t A B + t t b t v d t V B (6-22) With the definitions given
above this postulate reads d dt t 1 2 t v t ( ) v +t u d t V B = t ST ( ) n
t v d t A B + t t b t v d t V B (6-23) which can be transferred in a local
formulation using CAUCHYs eq. (6
complies with the original LSA acceptance test standards, and is in
condition for safe operation. Proper documentation of this maintenance
activity is required to be entered in the LSA records, and is also defined
by the manufacturer. Task specific traini
conservation of mass and balance of linear and angular momentum in
the local form. They provide seven partial differential equations, which
hold at every point of a continuum and for all time. As the initial density,
0 , and the body force, t b, are known
materials) and distinguish one material from the other are called
constitutive equations. They will be treated in chapter 7. But before, two
other universal principles used in continuum mechanics will be
addressed, which can be derived from the 33 Assumin
t 0 t 1 dt , (6-32) where the first bracket vanishes due to eq. (6-27), so
that the variational problem finally writes as I = i F xi d dt F x i t 0 t 1
dt = 0 . (6-33) As i (t) are arbitrary (test functions), the term written in
brackets has to vanish in
circle, which is referred to as MOHRs circle 30, is the locus of the
components of all possible stress vectors in a material point X, acting on
area elements under varying orientation. The stress components in the
actual coordinate system, xx , yy , ( ) x
function with f(1) = 0 and f '(1) = 1. The most common strain tensors are
EngMech-Script.doc, 29.11.2005 - 23 - BIOT'S 17 (linear) strain
tensor, f (i) = i 1 = 0 t i 0 t E(B) = 0 t U I = 1 2 0 0 t u + 0 0 t ( ) u
T , (4-40) GREEN-LAGRANGEan (quadratic) st
in the description of motion given by eq. (4-10), attention is given to a
point in space, and we study what is happening at that point as time
passes. This description is called the spatial description, and the
independent variables (t x, t) present in eq
decomposed into a spherical part and a deviatoric part, S , S = 1 3 ( ) trS
1 + S = p1 + S , (5-21) where p = 1 3 kk = hyd , (5-22) is the mean
pressure or (negative) hydrostatic stress. Deviatoric stresses play an
important role for plastic behaviour of
to some strain tensor, E(*) , if the stress power density in the reference
configuration is 0 win = T(*) E (*) = J t S t D , (5-33) see eq. (6-19).
According to this definition, 0 T is work conjugate to 0 t E(G) . For
CAUCHY's stress tensor there is no wo