5. stress and heat flux jan 2013
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Transcript of 5. stress and heat flux jan 2013
Theory of Stress & Heat Flux Stress: Scalar, Vector & Tensor
Cauchy Stress Principle, Conjugate Stress Tensors Fourier-Stokes Heat Flux Theorem
What is โStressโ?
Stress is a measure of force intensity either within or on the bounding surface of a body subjected to loads.
The Continuum Model takes a macroscopic approach:
Measurable aggregate behavior rather than the microscopic, atomistic activities that may in fact have led to them, and consequently, the
Standard results of calculus applicable in the case of limiting values of this quotient as the areas to which the forces are applied become very small.
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Stress
The simple answer of force per unit area raises the following questions:
What force?
As the size and shape of the material in question changes as motion evolves: What area:
What direction? Which surface? Which location?
A more rigorous definition of stress is required to settle these and other matters of importance.
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What is Stress?
In defining stress, care must be taken to note that sometimes we are talking about a
Tensor โ the Stress Tensor.
This completely characterizes the stress state at a particular location. That such a tensor exists is proved as Cauchyโs theorem โ a fundamental law in Continuum Mechanics.
Vector - the Traction โ Stress Vector
Intensity of resultant forces on a particular surface in a specific direction. This is, roughly speaking, what we have in mind when we say that stress is โforce per unit area.โ
Scalar โ the scalar magnitude of the traction vector or some other scalar function of the stress tensor.
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Three Stresses
โโฆ a distinction is established between two types of forces which we have called โbody forcesโ and โsurface tractionsโ, the former being conceived as due to a direct action at a distance, and the latter to contact action.โ AEH Love
It is convenient to examine these forces by categorizing them as follows:
Body forces ๐ (force per unit mass);
These are forces originating from sources (fields of force usually) outside of the body that act on the volume (or mass) of the body.
Surface forces i.e.: ๐(force per unit area of surface across they which they act)
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Two Forces
Body forces ๐ (force per unit mass); These are forces originating from sources - usually outside of the body
Fields of force Act on the volume (or mass) of the body.
These forces arise from the placement of the body in force fields Definition: A field of force is a Euclidean Point Space in which a
force function is specified at every point Examples: gravitational, electrical, magnetic or inertia
As the mass of a continuous body is assumed to be continuously distributed, any force originating from the mass is also continuously distributed.
Body forces are therefore assumed to be continuously distributed over the entire volume of the material.
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Body Forces
Force per unit area of surface across they which they act and are distributed in some fashion over a surface element of the body
Element could be part of the bounding surface, or an arbitrary element of surface within the body;
Examples: Shear stresses, normal stresses such as hydrostatic pressure, Wind loading, contact with another solid etc.
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Surface Forces
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Forces on an element ๐ surrounding a point ๐ ๐ฅ1, ๐ฅ2, ๐ฅ3 in a body acted upon by the forces ๐ 1, ๐ 2, โฆ , ๐ ๐.
The resultant body force per unit volume is ๐ ๐ฅ1, ๐ฅ2, ๐ฅ3 .
Consider an infinitesimal area element oriented in such a way that the unit outward normal to its surface is ๐ง ๐ฅ1, ๐ฅ2, ๐ฅ3 .
If the resultant force on the surface ฮ๐ of ฮ๐ is ฮ๐ญ, and this results in a traction intensity which will in general vary over ฮ๐.
Surface traction vector on this elemental surface be ๐ป, it is convenient to label this traction
๐ป ๐ . The superscript ๐ emphasises the fact that this traction is the resultant on the surface whose outward normal is ๐.
Forces on an Element
We can write that,
๐ป ๐ = limฮ๐โ0
ฮ๐ญ
ฮ๐=๐๐ญ
๐๐
IMPORTANT NOTE: The traction ๐ป ๐ gets its direction, not from ๐ but from ๐ญ. ๐ signifies the orientation of the surface on which it acts.
Or, conversely that
ฮ๐ญ = ๐ป ๐ ๐๐ฮ๐
In general, ๐ป ๐ = ๐ป ๐ ๐ฅ1, ๐ฅ2, ๐ฅ3 and ๐ = ๐ ๐ฅ1, ๐ฅ2, ๐ฅ3 as the surface itself is not necessarily a plane. It is only as the limit is approached that ๐ is a fixed direction for the elemental area and ฮ๐ญ and ๐ป ๐ are in the same direction.
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Surface Forces on an Element
Furthermore, the body force per unit mass is ๐๐. The density ๐ = ๐ ๐ฅ1, ๐ฅ2, ๐ฅ3 as it varies over the whole body. If the resultant body force in the volume element ฮ๐ is ฮ๐ญ, we can compute the body force per unit volume
๐ =1
๐limฮ๐โ0
ฮ๐ท
ฮ๐=1
๐
๐๐ท
๐๐
so that the body force on the element of volume
ฮ๐ท = ๐๐๐๐ฮ๐
.
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Body Forces on an Element
Vector intensity of the vector force on the surface as the surface area approaches a limit.
It is defined for a specific surface with an orientation defined by the outward normal ๐.
This implies immediately that the traction at a given point is dependent upon the orientation of the surface. It is a vector that has different values at the same point depending upon the orientation of the surface we are looking at.
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Surface Traction
It is a function of the coordinate variables.
It is therefore proper to write,
๐ ๐ง = ๐ ๐ง, ๐ฅ1, ๐ฅ2, ๐ฅ3 โก ๐ ๐ง ๐ฅ1, ๐ฅ2, ๐ฅ3
to make these dependencies explicitly obvious
In general, ๐ and ๐ง are not in the same direction; that is, there is an angular orientation between the resultant stress vector and the surface outward normal.
Surface Traction is expressed as the vector sum of its
projection ๐ก๐ โก ๐๐ง โ ๐ง along the normal ๐ง and
๐ก๐ โก ๐ ๐ง โ ๐ก๐๐ง on the surface itself.
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Surface Traction
It is easy to show that this shear stress vector is the
surface projection of the resultant ๐ โ ๐งโ ๐ง ๐ ๐ง . These normal and shearing components of the stress vector are called the normal and shear tractions respectively. They are the scalar magnitudes of the normal and tangential components of the reaction on any surface of interest.
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Normal & Shearing Stresses
The EulerโCauchy stress principle states that upon any surface (real or imaginary) that divides the body, the action of one part of the body on the other is equipollent to the system of distributed forces and couples on the surface dividing the body, and it is
represented by a vector field๐ ๐ง . In view of Newtonโs third law of action and reaction, this principle can be expressed compactly in the equation,
๐ โ๐ง = โ๐ ๐ง
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Euler-Cauchy Stress Principle
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Provided the stress vector ๐ ๐ง acting on a surface with outwardly drawn unit normal ๐ง is a continuous function of the coordinate variables, there exists a second-order tensor field ๐(๐ฑ), independent of ๐ง, such that ๐ ๐ง is a linear function of ๐ง such that:
๐ ๐ง = ๐๐ง
The tensor ๐ in the above relationship is the tensor of proportionality and it is called Cauchy Stress Tensor. It is also the โtrue stressโ tensor for reasons that will become clear later.
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Cauchyโs Theorem
To prove this expression, consider a tetrahedron with three faces oriented in the coordinate planes, and with an infinitesimal base area ๐๐ด oriented in an arbitrary direction specified by a normal vector ๐ง (Figure 6.3). The tetrahedron is formed by slicing the infinitesimal element along an arbitrary plane ๐ง. The stress vector on this plane is denoted by ๐ ๐ง . The stress vectors acting on the faces of the tetrahedron are denoted as ๐ป ๐1 , ๐ป ๐2 and ๐ป ๐3 From equilibrium of forces, Newtonโs second law of motion, we have
๐โ
3๐๐ด ๐ = ๐ ๐ง ๐๐ด โ ๐ ๐1 ๐๐ด1 โ ๐
๐2 ๐๐ด2 โ ๐๐3 ๐๐ด3
= ๐ ๐ง ๐๐ด โ ๐ ๐i ๐๐ด๐
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Cauchy Stress Theorem
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where the right-hand-side of the equation represents the product of the mass enclosed by the tetrahedron and its acceleration: ๐ is the density, ๐ is the acceleration, and โ is the height of the tetrahedron, considering the plane ๐ as the base.
๐โ
3๐๐ด ๐ = ๐ป ๐ ๐๐ด โ ๐ป ๐1 ๐๐ด1 โ ๐ป
๐2 ๐๐ด2 โ ๐ป๐3 ๐๐ด3
= ๐ป ๐ ๐๐ด โ ๐ป ๐๐ ๐๐ด๐
The area of the faces of the tetrahedron perpendicular to the axes can be found by projecting ๐๐ด into each face:
๐๐ด๐ = ๐ โ ๐๐ ๐๐ด = ๐๐๐๐ด
and then substituting into the equation to cancel out ๐๐ด:
๐ ๐ง ๐๐ด โ ๐ ๐๐ ๐๐๐๐ด = ๐โ
3๐๐ด ๐
To consider the limiting case as the tetrahedron shrinks to a point, โ โ 0, that is the height of the tetrahedron approaches zero. As a result, the right-hand-side of the equation approaches 0, so the equation becomes,
๐ ๐ง = ๐ป ๐๐ ๐๐
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Cauchy Theorem
We are now to interpret the components ๐ป ๐๐ in this
equation. Consider ๐ป ๐1 the value of the resultant stress traction on the first coordinate plane. Resolving this along the coordinate axes, we have,
๐ป ๐1 = ๐1 โ ๐ป๐1 ๐1 + ๐2 โ ๐ป
๐1 ๐2 + ๐3 โ ๐ป๐1 ๐3
= ๐11๐1 + ๐12๐2 + ๐13๐3 = ๐1๐๐๐
Where the scalar quantity ๐1๐is defined by the above
equation as,
๐1๐ = ๐๐ โ ๐ป๐1
or in general, we write the components as,
๐๐๐ = ๐๐ โ ๐ป๐๐ , ๐ = 1,2,3
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Interpretation
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Above figure is a graphical depiction of this definition
where we can see that ๐๐๐ = ๐๐ โ ๐ป๐๐ is the scalar
component of the stress vector on the ๐ coordinate plane in the ๐ direction. For any coordinate plane therefore, we
may write, ๐ป ๐๐ = ๐๐๐๐๐, so that the stress or traction
vector on an arbitrary plane determined by its orientation in the outward normal ๐,
๐ป ๐ = ๐ป ๐๐ ๐๐ = ๐๐๐๐๐๐๐
Which is another way of saying that the component of the vector ๐ป ๐ along the ๐ coordinate direction is ๐๐๐๐๐ which
is the contraction, ๐ ๐ฑ, ๐ก ๐ง = ๐ป ๐ง . This proves Cauchy Theorem.
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Interpretation
Obviously, ๐๐๐ are the components of the stress tensor in
the coordinate system of computation that we have used so far. The Cauchy law, being a vector equation remains valid in all coordinate systems. We will then have to compute the different values of the stress tensor in the system of choice when for any reason we choose to work in not-Cartesian coordinates.
Earlier on, we introduced the normal ๐ก๐ โก ๐ป๐ โ ๐ and
shearing ๐ป ๐ โ ๐ก๐๐ components of the stress vector. We can compute these values now in terms of the scalar components of the stress tensor. Using Cauchy theorem, we have that,
๐ = ๐ก๐ โก ๐ป๐ โ ๐ = ๐ ๐, ๐ก ๐ โ ๐ = ๐ โ ๐๐
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Normal Stress Again
This double contraction defines the scalar value which is the magnitude of the stress vector acting in the direction of the normal to the plane. The other important scalar quantity โ the magnitude of the corresponding projection of the traction vector to the surface itself is obtained from Pythagoras theorem:
๐ = ๐ป ๐ โ ๐ก๐๐ = ๐ป ๐ โ ๐๐
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Shear Stress
The arguments that proved Cauchy theorem could have been based on non-Cartesian coordinate systems. The stress equation must remain unchanged however and the stress tensor characterizing the state of stress at a point remains an invariant. As it is with any second-order tensor, its components in general coordinates will be obtained from,
๐(๐ฑ) = ๐๐๐๐ ๐โ๐ ๐= ๐๐๐๐ ๐โจ๐ ๐ = ๐.๐
๐ ๐ ๐โ๐ ๐= ๐๐.๐๐ ๐โจ๐ ๐
Because the Cauchy stress is based on areas in the deformed body, it is a spatial quantity and appropriate to an Eulerian kinematic formulation. Whenever it is more convenient to work in Lagrangian coordinates, a stress tensor based on this may become more appropriate.
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Tensor Bases
From last chapter, we recall that the vector current area in a deformed body ๐๐ = ๐ฝ๐๐จ โ ๐ญโ1 where ๐๐จ is its image in the material coordinates. The resultant force acting on an area bounded by ฮ๐ in the deformed coordinates can be obtained, using Cauchy stress theorem as,
๐๐ = ๐๐ฮ๐
โ ๐ = ๐ฝ ๐๐จ โ ๐ญโ1
ฮ๐0
โ ๐
= ๐๐จ โ ๐ฝ๐ญโ1
ฮ๐0
โ ๐
= ๐๐จ โ ฮ๐0
๐ต
where ๐ต โก ๐ฝ๐ญโ1๐ is called the Nominal Stress Tensor
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Nominal Stress
The transpose of the nominal stress is called the First Piola-Kirchhoff Stress Tensor ๐. Consequently,
๐ โก ๐ต๐ = ๐ฝ๐ โ ๐ญโ๐
This transformation applied to ๐ to produce ๐, when applied as in this or any other case to any tensor is called a Piola transformation.
In the above equation, we have used the yet-to be proved fact that the Cauchy stress tensor is symmetric. This will be established later.
The components of the ๐ stress are the forces acting on the deformed configuration, per unit undeformed area. They are thought of as acting on the undeformed solid.
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First Piola-Kirchhoff Stress
Recall that the ratio of elemental volumes in the spatial to material, coordinates
๐๐ฃ
๐๐= ๐ฝ = det ๐ญ
where ๐ญ is the deformation gradient of the transformation. It therefore follows that the Kirchhoff stress ๐, defined by
๐ = ๐ฝ๐
is no different from Cauchy Stress Tensor during isochoric (or volume-preserving) deformations and motions. It is used widely in numerical algorithms in metal plasticity (where there is no change in volume during plastic deformation).
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Kirchhoff Stress
The second Piola Kirchhoff Stress, ๐ต is useful in Conjugate stress analysis. It is defined by,
๐ต โก ๐ฝ๐ญโ1 โ ๐ = ๐ฉ โ ๐ญ๐ In terms of the Cauchy stress ๐ we can write,
๐ฉ = ๐ฝ๐ญโ1 โ ๐ โ ๐ญโ๐ . In terms of the nominal stress tensor, ๐ฉ = ๐ต โ ๐ญโ๐ . Note that the First Piola-Kirchhoff Stress Tensor is not
symmetric. On the other hand, the 2nd P-K tensor is symmetric just like the Cauchy tensor. Furthermore if the material rotates without a change in stress state (rigid rotation), the components of the 2nd Piola-Kirchhoff stress tensor will remain constant, irrespective of material orientation. It is also important as an energy conjugate to the Lagrange strain.
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Second Piola-Kirchhoff Stress
As a second order symmetric tensor, the Cauchy Stress Tensor has positive definite quadratic forms when operated on by real vectors. Its eigenvalues are real and its principal invariants are:
๐ผ1 ๐ = tr ๐ = ๐๐๐๐ ๐ โ ๐ ๐= ๐๐๐๐ ๐ โ ๐ ๐ = ๐.๐
๐ ๐ ๐ โ ๐ ๐=
๐๐.๐๐ ๐ โ ๐ ๐= ๐๐
.๐
๐ผ2 ๐ =tr(๐) 2โ tr ๐๐
2=1
2๐๐๐๐๐๐โ ๐๐
๐๐๐๐
(Half of the square of the trace of the tensor minus the trace of the square of the tensor ๐. The third and last scalar invariant of a tensor ๐ is its determinant if it exists.)
๐ผ3 ๐ = det ๐ = ๐๐๐๐๐๐1๐๐2๐๐3
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Extremal Values & Principal Planes
Because of its simplicity, working and thinking in the principal coordinate system is often very useful when considering the state of the elastic medium at a particular point. Principal stresses are often expressed in the following equation for evaluating stresses in the ๐ฅ and ๐ฆ directions or axial and bending stresses on a part. The principal normal stresses can then be used to calculate the Von Mises stress and ultimately the safety factor and margin of safety
๐1, ๐2 =๐๐ฅ + ๐๐ฆ
2ยฑ
๐๐ฅ โ ๐๐ฆ
2
2
+ ๐๐ฅ๐ฆ2
Using just the part of the equation under the square root is equal to the maximum and minimum shear stress for plus and minus. This is shown as:
๐1, ๐2 = ยฑ๐๐ฅ โ ๐๐ฆ
2
2
+ ๐๐ฅ๐ฆ2 .
We are now in a position to recalculate the normal and shearing components of the traction vector ๐ป ๐
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Stress Systems
๐ = ๐ โ ๐๐ = ๐1 ๐2 ๐3
๐1 0 00 ๐2 00 0 ๐3
๐1๐2๐3
= ๐1(๐1)2 + ๐2(๐2)
2 + ๐3(๐3)2
And the shear stress is the scalar magnitude of,
๐ป ๐ โ ๐ก๐๐ = ๐๐ โ ๐ โ ๐๐ = ๐ โ ๐ โ ๐ ๐
=
๐1 0 00 ๐2 00 0 ๐3
โ
๐1๐1 0 00 ๐2๐2 00 0 ๐3๐3
๐1๐2๐3
โด ๐
= (๐1๐1)2 + (๐2๐2)
2 + (๐3๐3)2 โ ๐1(๐1)
2 + ๐2(๐2)2 + ๐3(๐3)
2 2 From these, we can write in a more compact form that,
๐2 + ๐2 = (๐1๐1)2 + (๐2๐2)
2 + (๐3๐3)2
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The values of normal and shear stresses given in terms of the principal coordinates can be solved (using Mathematica) after noting that, (๐1)
2 + (๐2)2 + (๐3)
2 =0 as shown below to obtain the Mohr circle of stress in three dimensions.
The values of the square of the direction cosines follow the three circles (๐1)
2 โ ๐2 + ๐2 โ ๐๐3 + ๐2 ๐3 โ ๐ (๐2)2 โ ๐2 + ๐2 โ ๐๐3
+ ๐1 ๐3 โ ๐ (๐3)
2 โ ๐2 + ๐2 โ ๐๐2 + ๐1 ๐2 โ ๐
In the figure, we used the symbols, ๐ผ1 = (๐1)2, ๐ผ2 =
(๐2)2and ๐ผ3 = (๐3)
2 with principal stress values of ๐1 = 50, ๐2 = 15, ๐3 = 5.
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Mohr Circle
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We present here examples of states of stress as an illustration:
1. Hydrostatic Pressure
2. Uniaxial Tension
3. Equal Biaxial tension
4. Pure Shear
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States of Stress
๐ ๐ฑ = โ๐๐ = โ๐๐๐๐ ๐ ๐โ๐ ๐ = โ๐ ๐ ๐โ๐ ๐
In a Cartesian system, we have, ๐ ๐ฑ = โ๐๐ = โ๐ ๐๐โ๐๐ .
For a surface whose outward normal is the unit vector ๐ง, the traction
๐ ๐ง = ๐ ๐ฑ, ๐ก ๐ง = โ๐๐๐๐ ๐ ๐โ๐ ๐ ๐ง
= โ๐๐๐๐๐ ๐ ๐ง โ ๐ ๐ = โ๐๐๐๐๐๐๐ ๐
= โ๐๐๐๐ ๐ = โ๐๐ง
Furthermore, the scalar normal traction
๐ = ๐ ๐ง โ ๐ง = โ๐๐ง โ ๐ง = โ๐
And the shear stress: the magnitude of the vector difference between the traction and the vector normal traction.
๐ = ๐ ๐ง โ ๐๐ง = 0
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Hydrostatic Pressure
Define uniaxial tension as a state where there is a normal traction ๐ก in a given direction (unit vector ๐)and zero traction in directions perpendicular to it. The Cauchy stress for this is ๐ ๐ฑ = ๐ก ๐โ ๐ . The traction on a surface with unit vector ๐ง is,
๐ ๐ง = ๐ ๐ ๐ง = ๐ก ๐โ ๐ ๐ = ๐ก๐ ๐ โ ๐ = ๐ก๐ cos ๐
From which we can see that, under uniaxial stress, the traction is always directed along the vector ๐ no matter what the orientation of the surface might be.
Of course, when ๐ = ๐/2, traction is zero.
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Uniaxial Tension
Consider the stress tensor, ๐ ๐ฑ = ๐ก ๐โ ๐ + ๐โ ๐ where ๐ and ๐ are perpendicular directions. The traction on an arbitrary plane ๐ง
๐ ๐ง = ๐ ๐ ๐ง = ๐ก ๐โ ๐ + ๐โ ๐ ๐ง = ๐ก๐ ๐ง โ ๐ + ๐ก๐ ๐ง โ ๐ = ๐ก๐ cos ๐ + ๐ก๐sin๐
The eigenvalues of ๐ ๐ฑ are ๐ก, ๐ก, 0 . ๐ ๐ฑ this case has the spectral form,
๐ ๐ฑ = ๐ก๐ฎ1โ๐ฎ1 + ๐ก๐ฎ2โ๐ฎ2
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Biaxial Traction
Consider the stress tensor, ๐ ๐ฑ = ๐ก ๐โ ๐ + ๐โ ๐ where ๐ and ๐ are perpendicular directions. The traction on an arbitrary plane ๐ง
๐ ๐ง = ๐ ๐ ๐ง = ๐ก ๐โ ๐ + ๐โ๐ ๐ง = ๐ก๐ ๐ง โ ๐ + ๐ก๐ ๐ง โ ๐ = ๐ก๐ sin ๐ + ๐ก๐cos๐
The eigenvalues of ๐ ๐ฑ are ๐ก, โ๐ก, 0 . Furthermore, ๐ in this case has the spectral form,
๐ ๐ฑ = ๐ก๐ฎ1โ๐ฎ1 โ ๐ก๐ฎ2โ๐ฎ2
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Pure Shear
Theory of Heat Fluxes Fourier Stokes Heat Flux Theorem
Cauchyโs postulated the existence of a stress tensor on the basis of which the load intensity arising from mechanical forces (body and surface forces) can be elegantly quantified in a consistent manner.
The counterpart of this for thermal exchanges with the surroundings is the Fourier-Stokes heat flux theorem.
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Heat Fluxes
Consider a spatial volume B๐ก with boundary ๐B๐ก. Let the outwardly drawn normal to the surface be the unit vector n. Fourier Stokes heat Flux Principle states that
โ๐ช ๐ฑ, ๐ก โ vector field such that, the heat flow out of the volume is
โ ๐ฑ, ๐ก, ๐B๐ก = โ ๐ฑ, ๐ก, ๐ง = โ๐ ๐ฑ, ๐ก โ ๐ง
๐ ๐ฑ, ๐ก is called the heat flux through the surface.
[email protected] 12/30/2012 Department of Systems Engineering, University of Lagos 42
Fourier-Stokes Theorem
Heat flow into the spatial volume B๐ก volume is
๐ ๐ฑ, ๐ก โ ๐ง๐B๐ก
๐๐ = ๐ฝ๐ ๐ฑ, ๐ก โ ๐ โT๐๐B
๐๐ด
= ๐ฝ๐ โ1๐ ๐ฑ, ๐ก โ ๐B
๐๐จ = ๐ โ ๐B
๐๐จ
๐ ๐ฑ, ๐ก is a Piola transformation of the spatial heat flux. That is,
๐ ๐ฟ, ๐ก = ๐ฝ๐ โ1๐ ๐ฑ, ๐ก
[email protected] 12/30/2012 Department of Systems Engineering, University of Lagos 43
Fourier-Stokes Theorem