Page 161 - The Combined Finite-Discrete Element Method

144 DEFORMABILITY OF DISCRETE ELEMENTS

• T is symmetric and positive, i.e. for any vector a
a·Ta > 0 unless a = 0 (4.63)

• T satisfies the equation of motion
divT + b = ρ˙v (4.64)

where s is the traction force corresponding to the surface of deformed configuration, b is
the body force per unit volume of the deformed configuration, ρ is the density measured
per unit volume of the deformed configuration, and m is the normal to the boundary of
deformed configuration.
The Cauchy stress tensor in essence represents a linear mapping where a given outward
surface normal m is mapped onto a total surface traction force s. The surface normal m is
of magnitude equal to the surface area it represents. Thus, for instance, if the magnitude of
m is doubled, the total surface traction is doubled. This is easily understood, for doubling
the normal m is equivalent to doubling the surface area.
The matrix of Cauchy stress tensor in the global frame

(i, j, k) (4.65)

is given by
 
t xx t xy t xz
T =  t yx t yy t yz  (4.66)
t zx t zy t zz
where the first index indicates the direction of the stress component (direction of traction
force) and the second index denotes the corresponding surface normal. Thus, t xy is the
traction force in the x-direction on the surface ‘in the y-direction’, i.e.

force t xy i on the surface 1j (4.67)

Cauchy stress refers to the force per unit area of the deformed configuration. Components
of Cauchy stress tensor are shown in Figure 4.7.
For any given surface defined by surface normal m (Figure 4.8), the surface traction
in global orthonormal frame
(i, j, k) (4.68)

is obtained by simply multiplying the matrix of tensor T with the matrix of vector m,i.e.

     
s x t xx t xy t xz m x
s = s x i + s y j + s z k =  s y  =  t yx t yy t yz   m y  (4.69)
s z t zx t zy t zz m z
From this expression, it is obvious that the Cauchy stress is a linear mapping from the
space of normals (surfaces) into a space of forces, where each normal is mapped onto a
corresponding traction force. This is logical, because in mathematical terms a tensor is