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Cartesian Tensors and Solid Mechanics
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Introduction Continuum Mechanics
Continuum Mechanics is the branch of mechanics used to investigate the deformation and
flow of materials subjected to loads.
Is a generalization of the classical Newtonian mechanics to macroscopic bodies.
These bodies are considered formed by infinite collections of material points.
As in classical mechanics, the fundamental physical notion is the displacement of material
points in the body as a result of applied external forces.
The key new concepts are stress and strain.
The mathematical formulation of problems in continuum mechanics involves two sets of
equations.
The first set of governing equations consists of conservation principles of universal validity.
The second set are empirical, material-specific relationships called constitutive equations
of behavior.
The full set of equations often intimidate the beginning student.
The use of Cartesian tensor notation significantly simplifies the equations and also helps
gaining insight and understanding of the key concepts involved.
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The Equations of Continuum Mechanics
Introduce a three dimensional Cartesian system of coordinates and consider a material body
subjected to the action of externally applied loads.
2.1
Principle of Mass Conservation
The principle of mass conservations states that mass is never created nor destroyed.
2.2
Principle of Momentum Conservation
The principle of momentum conservation is the generalization of Newton’s second law of
motion to continuous media and it is written, for any point in the body as
ρ
∂ 2 ux
∂σxx ∂σxy ∂σxz
+
+
+ ρfx
=
2
∂t
∂x
∂y
∂z
ρ
∂σyx ∂σyy ∂σyz
∂ 2 uy
=
+
+
+ ρfy
2
∂t
∂x
∂y
∂z
ρ
∂ 2 uz
∂σzx ∂σzy ∂σzz
+
+
+ ρfz
=
2
∂t
∂x
∂y
∂z
where ux , uy , uz are the components of the displacement vector at the point.
The term on the left hand side is the product of mass times acceleration.
The first three terms on the right hand side are the mutual interaction forces among
particles of the body. Specifically, σxy is the component of the stress tensor representing
force along the x direction acting on a plane whose normal is y and x .
All other stress tensor components are defined similarly.
The last term are the components ρfx , ρfy , ρfz of volume forces acting on the body (e.g.
gravity, electromagnetic).
The above set of equations are often referred as the equation of motion of the continuous
body.
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2.3
Constitutive Equations (Hooke’s law)
For illustration purposes, a specific set of constitutive equations are examined.
These are expressions relating stress to strain and describing the linear elastic behavior
of isotropic bodies.
They generalize the emprirical notion that the as the stretch goes the force.
They were first discovered by Hooke.
σxx = λ(ǫxx + ǫyy + ǫzz ) + 2Gǫxx = λΘ + 2Gǫxx
σyy = λ(ǫxx + ǫyy + ǫzz ) + 2Gǫyy = λΘ + 2Gǫyy
σzz = λ(ǫxx + ǫyy + ǫzz ) + 2Gǫzz = λΘ + 2Gǫyy
σxy = 2Gǫxy
σyz = 2Gǫyz
σzx = 2Gǫzx
Lambda and G are Lame’s material specific elastic constants.
When Hooke’s law is combined with the equation of motion one obtains Navier’s equations.
Navier’s equation’s are the starting point of approximative finite element modeling computations.
2.4
Navier’s Equations
∂Θ
∂ 2 ux ∂ 2 ux ∂ 2 ux
∂ 2 ux
=
(λ
+
G)
+
G(
+
+
) + ρfx
∂t2
∂x
∂x2
∂y 2
∂z 2
∂ 2 uy
∂Θ
∂ 2 uy ∂ 2 uy ∂ 2 uy
ρ 2 = (λ + G)
+ G( 2 +
+
) + ρfy
∂t
∂y
∂x
∂y 2
∂z 2
∂Θ
∂ 2 uz ∂ 2 uz ∂ 2 uz
∂ 2 uz
+ G( 2 +
+
) + ρfz
ρ 2 = (λ + G)
∂t
∂z
∂x
∂y 2
∂z 2
ρ
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Cartesian Tensor Notation
Cartesian tensor notation yields the governing equations in its simplest form.
The notation also sheds light and insight on the significance of the various terms and
quantities involved.
The notation is simply based on the clever use of indices and a couple of notational
conventions.
Specifically, the index notation is
(x, y, z) → (x1 , x2, x3)
(i, j, k) → (e1 , e2, e3)
The notational conventions include:
The summation convention that repeated indices in a term imply summation over the
range of the indices.
The comma convention that when an index is preceeded by a comma partial derivative
with respect to the corresponding indexed coordinate is implied.
Using Cartesian tensor notation, the position vector is written as
xi + yj + zk = x1 e1 + x2 e2 + x3e3 =
3
X
xi ei = xi ei = r
i=1
The displacement vector is written as
ux i + uy j + uz k = u1 e 1 + u2 e 2 + u3 e 3 =
3
X
ui e i = ui e i = u
3
X
vi ei = vi ei = v
i=1
And the velocity vector is written as
vx i + vy j + vz k = v1 e1 + v2 e2 + v3 e3 =
i=1
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Note the use of indices and the summation convention in these expressions.
The writing of more complex entities is simplified and makes their definitions easier to
understand, memorize and remember.
For instance, the tensor formed by the partial derivatives of the displacements, called the
deformation gradient tensor is



∂u1
∂x1
∂u1
∂x2
∂u1
∂x3
∂u2
∂x1
∂u2
∂x2
∂u2
∂x3
∂u3
∂x1
∂u3
∂x2
∂u3
∂x3



=
∂ui
= ui,j
∂xj
Note the use of indices and the comma notation in this expression.
Strain is the relative amount of elongation. The strain tensor is defined as




∂ux
∂x
1 ∂uy
(
2 ∂x
1 ∂uz
(
2 ∂x
+
+
∂ux
)
∂y
∂ux
)
∂z
1 ∂ux
(
+
2 ∂y
∂uy
∂y
1 ∂uz
(
+
2 ∂y
∂uy
)
∂x
∂uy
)
∂z
1 ∂ux
(
+
2 ∂z
1 ∂uy
(
+
2 ∂z
∂uz
∂z
1 ∂ui ∂uj
+
)=
= (
2 ∂xj
∂xi

ǫxx

=  ǫyx
ǫzx
∂uz
)
∂x
∂uz
)
∂y




=
1
(ui,j + uj,i ) =
2

ǫxy ǫxz
ǫyy ǫyz 
 = ǫij
ǫzy ǫzz
Note again the use of indices and the comma notations in this expression.
The strain can be regarded as composed of two terms: strain without volume change and
strain without shear
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ǫij = ǫ′ij + δij Θ
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The first term is called the strain deviator tensor and the second is the dilation given as
ǫxx + ǫyy + ǫzz = tr(ǫij ) = Θ
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δij is the Kronecker delta (= 1 if i = j, 0 otherwise).
Stress is the result of the interaction of material particles. The stress tensor is defined as


σxx σxy σxz


 σyx σyy σyz  = σij
σzx σzy σzz
As with the strain, the stress can also be regarded as composed of two terms:
σij = σij′ + δij σm
The first term is the stress deviator tensor and it plays a key role in nonlinear mechanics.
The second term is the mean stress defined as
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1
(σxx + σyy + σzz ) = tr(σij ) = σm
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The long expression describing the vector of spatial derivatives of the stress tensor is
reduced to a very simple form when Cartesian tensor notation is used
σij,j
Note the power of the use of indices, the summation convention and the comma notation.
The set of equations describing Hooke’s law is reduced to a single, easily remembered
expression
σij = λǫii δij + 2Gǫij
The equation of motion is reduced to a single easily remembered expression.
ρüi = σij,j + ρfi
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Finally, Navier’s equations are also reduced to a very compact form.
ρüi = (λ + G)uj,ji + Gui,jj + ρfi
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Summary
The simplifying power of Cartesian tensor notation has been demonstrated using as example
the equations of solid mechanics.
The notation also gives insight into the significance of the various terms and quantities
involved in the equations
and makes them easy to remember and manipulate.
σij = λǫii δij + 2Gǫij
ρüi = σij,j + ρfi
ρüi = (λ + G)uj,ji + Gui,jj + ρfi
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