ij ei e j Vector Product (Cross Product) two vectors: w = u v = ui vj ei e j
= ijkui vj ek = v u LEVI-CIVITA symbol (permutation symbol): ijk = 1
if i, j, k take values in cyclic order 1 if i, j, k take values in acyclic order
0 if two of i, j, k take the
0 t F , (4-78) and is obviously not objective. From this, the
transformations of the other kinematical quantities follow, 0 t R =t Q0 t
R rotation tensor 0 t U = 0 t U right stretch tensor, 0 t E (B) = 0 t E(B)
BIOT's strain tensor, 0 t E (G) = 0 t E(G) G
i.e. body forces and mass accelerations, are small of higher order
compared to the surface integrals, that is ti dAi i=1 3 + t(n) dAn = t(ei )
dAi i=1 3 + t(n) dAn = o . (5-7) As by eq. (5-6), the fourth normal, n,
can be expressed by the first three, ni
= 1 2 [ ] TiiTkk TikTki IIIT = detT = ijkTi1 Tj 2Tk 3 = ijkT1i T2 j T3k .
Every linear combination of the three principal invariants is again
invariant against a coordinate transformation. T and TT have the same
invariants. A2.4 Eigenvalues and Eigenvecto
42. (8-3b,c) These equations hold for all material points of an arbitrary
body. They have to be completed by boundary conditions, specifying the
particular geometry and loading. Together, they form the boundary value
problem for the specific component. Ge
acceleration transforms as an objective vector, t a = t Q t a . Such special
transformations are called GALILEI transformation, characterised by t c
= o and t Q = 0 , i.e. t = o . The velocity gradient transforms like t L = t
gradt v = t t ( ) v T = 0 t F
R0 has been assumed for simplicity, since R is small. Depending on R,
the stresses in the joint cause elastic or elasto-plastic deformations. It is
assumed, that the deformations are elastic. The stress state is
axisymmetric, r = 0, so that the static pro
length remains undetermined. They can hence be normalised vi v j = ij =
1 for i = j 0 for i j Eigenvectors form an orthogonal and normalised
system of base vectors with respect to which the matrix of T is diagonal,
T = 1 0 0 0 2 0 0 0 3 ( ) vi v j = 1 v1v
Reference Frame If two observers describe their spaces by position
vectors with respect to their individual points of reference, then the
first observer sees the position vector of a spatial point X E3 at time t
with respect to his point of reference ("or
with respect to time, we obtain the acceleration t a =t v =t x = 2 t ( 0 x,t)
t 2 0 x , (4-84) EngMech-Script.doc, 29.11.2005 - 29 - and under the
change of reference frame t a =t x =t t x t ( ) c + t t x t ( ) c +t Q t v +t
Q t v +t c =t t x t ( ) c + t
law of motion, eq. (6-13). Eq. (7-4) implies that eq. (7-1) takes the form
t S(X ) = f =t cfw_ (Y, ) (X, ) X,Y B . (7-5) It is known from experience
that the stresses in a material point do not depend on the motion of other
points, provided that those ar
copy is inversely proportional to its importance. MURPHYs Law of the
Open Road When there is a very long road upon which there is a oneway bridge placed at random, and there are only two cars on that road, it
follows that: (1) the two cars are going in op
to decide which quantities to use as independent variables and which as
dependent variables. It has become more or less common practice to
consider the stresses as dependent variables, and motions as independent
ones. The whole history of motion may affec
points up this relation more clearly, t D = 1 2 t gij t gi tg j , t E (G) = 1 2
t gij 0 gi 0 g j . (4-67) The components of t D and t E (G) are equal,
namely half the material time derivatives of the covariant time metric
coefficients in t B, but the base
P2, generated by the imaginary cutting, the respective exterior normals
of every surface element at the interface are opposite to each other, n1 +
n2 = 0 . According to NEWTONs third law of motion, see chapter 6, it
is postulated that the corresponding su
e1 + cos e2 e3 = e3 . The components of a vector and a tensor v = vi ei
= vi ei , T = Tij ei e j = Tij ei e j . transform like vi = aij vj or vi =
aijvj = ajivj Tij = aikajl Tkl or Tij = aik ajlTkl = akialjTkl . EngMechScript.doc, 29.11.2005 - 72 - A1.4 V
stress tensor has been shown to be objective. eq. (5-22). It is also
necessary to establish a rule for the observer dependence of the
constitutive equations. Principle of Material Objectivity: The stress
power is objective (and thus also invariant) under
or = 0.5. In this case, hydrostatic stresses can not be determined from the
strains, see eq. (7-12). The range of physically possible values of
POISSON's ratio is hence 0 0.5 . 38 GABRIEL LAM (1795-1870) 39
THOMAS YOUNG (1773-1829) 40 S.D. Poisson (1781-1
E(G) = 1 2 0 t ( ) C I , can be derived. In the following, we shall
consider the simplest case of a linear relation between CAUCHY's stress
tensor, t S , and the linear strain tensor, E, known as HOOKE's law of
elasticity for small deformations. 7.2 Linea
we obtain t S(X ) = f = t F(X ), cfw_ ( ) F(X ) ,. . (7-4) If we take the
neighbourhood as arbitrarily small, 0, we obtain the material functional
of so-called simple materials t S(X ) = f = t cfw_ F(X ) . (7-5) Additional
principles and assumptions all
p , (8-64) with Ri = R t 2 , Ro = R + t 2 . Axial stresses have to obey eq.
(8-56)1 and (8-57)2. As c1 may now depend on r, the integration of (857)2 requires an assumption on 1(r). Commonly, 1 r = 0 is assumed as
above, which results in 11 = Ri 2 Ro 2 Ri
body B occupying a configuration t B of volume t V at time t is t f(B) =t
fb (B) +t fc (B) . (5-1) All vector fields are assumed to be objective, t f =
Q t f . (5-2) Body forces are forces that act on every element dB B and
hence on the entire volume of t
We can now apply the EUCLIDean transformation to the kinematic
quantities describing motion and deformation of a body. The motion of a
body, t x = t ( ) X,t , is described by identifying its materials points, X,
by their placement in a reference configura
= 11 E , 22 = 33 = 11 . In this way, YOUNGs modulus and POISSONs
ratio of an elastic material can be determined from a uniaxial tensile test
(see section 8.3). Other elastic constants used are the shear modulus G,
relating shear stresses and strains by
EngMech-Script.doc, 29.11.2005 - 48 - 7. Constitutive Equations All
foregoing axioms, laws, equations are supposed to be valid for all
material bodies and are thus labelled universal. In contrast, material
theory aims at describing the individual behaviou
with the physical dimension force per unit area, which is referred to as
surface force density or stress vector or traction. It depends on the locus
and the orientation of the surface element, which we describe by the
position vector, t x, and the exterio