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Chapter 2
Governing Equations
2.2 Mathematical description of internal forces in solids Our next objective is to outline the mathematical formulas that describe internal and external forces acting on a solid. Just as there are many different strain measures, there are several different definitions of internal force. We shall see that internal forces can be described as a second order tensor, which must be symmetric. Thus, internal forces can always be quantified by a set of six numbers, and the various different definitions are all equivalent.
2.2.1 Surface traction and internal body force
Forces can be applied to a solid body in two ways.Â
(i) A force can be applied to its boundary: examples include fluid pressure, wind loading, or forces arising from contact with another solid.Â
(ii) The solid can be subjected to body forces, which act on the interior of the solid. Examples include gravitational loading, or electromagnetic forces.
Formally,
let dA be an element of area on a
surface. Suppose that dA is subjected to a force
The resultant force acting on any portion S of the surface of the deformed solid is
Surface traction, like `true stress,’ should be thought of as acting on the deformed solid.
The traction vector is often resolved into components acting normal and tangential to a surface, as shown in the picture.Â
The normal component is referred to as the normal traction, and the tangential component is known as the shear traction.
Formally, let n denote a unit vector normal to the surface. Then
Formally, let dV
denote an infinitesimal volume element within the deformed solid, and let
The resultant body force acting on any volume VÂ within the deformed solid is
2.2.2 Traction acting on planes within a solid
Every plane in the interior of a solid is subjected to a distribution of traction. To see this, consider a loaded, solid, body in static equilibrium. Imagine cutting the solid in two. The two parts of the solid must each be in static equilibrium. This is possible only if forces act on the planes that were created by the cut.
Formally,
let dA be an element of area in the
interior of the solid, with normal n. Suppose that the material on the underside
of dA is subjected to a force
Note that internal traction is the force per unit area of the deformed solid, like `true stress’
The first term is the resultant force acting on the internal surface A, the second term is the resultant body force acting on the interior V.
To see this, note that the forces acting on planes separating two adjacent volume elements in a solid must be equal and opposite.
Let
where
To see this, consider the forces acting on the infinitessimal
tetrahedron shown in the figure. The
base and sides of the tetrahedron have normals in the
Recall
that
Finally,
let
so
or, using index notation
The
significance of this result is that the tractions acting on planes with
normals in the
2.2.3 The Cauchy (true) stress tensor
Consider
a solid which deforms under external loading. Let
Then, the tractionÂ
To see this, recall the last result from the preceding section
and substitute for
The
Cauchy stress tensor completely characterizes the internal forces acting in a
deformed solid. The physical
significance of the components of the stress tensor is illustrated in the figure:
Note the Cauchy stress represents force per unit area of the deformed solid. In elementary strength of materials courses it is called `true stress,’ for this reason.
HEALTH WARNING: Some texts define stress as the transpose of the definition used here,
so that
2.2.4 Other stress
measures
Cauchy
stress
The
other stress measures regard forces as acting on the undeformed solid.Â
Consequently, to define them we must know not only what the deformed
solid looks like, but also what it looked like before deformation. The deformation is described by a
displacement vector
as outlined in Section
2.1. In addition, let
We then define the following stress measures
The inverse relations are
also useful
The Kirchoff stress has no obvious physical significance.Â
The nominal stress tensor can be regarded as the internal force per unit undeformed area acting within a solid, as follows 1. Visualize an element of area dA in the deformed solid, with normal n, which is subjected to a force 2. Suppose that the element of area dA has started out as an element of
area 3. Then, the force
To see this, note that one can show (see Appendix D) that
Recall that the Cauchy stress is defined so that
Substituting
for
The material stress tensor can also be visualized as force per unit undeformed area, except that the forces are regarded as acting within the undeformed solid, rather than on the deformed solid. Specifically 1. The infinitesimal force 2. This means that we can define a (fictitious) force
in the reference configuration 3. This fictitious force is related to material stress
by Â
To see this, substitute
into the expression relating
Finally multiply through
by
where we have noted that
In
practice, it is best not to try to attach too much physical significance to
these stress measures. Cauchy stress
is the best physical measure of internal force
Specifically,
we shall show later that the rate of work
where
2.2.5 Stress measures for infinitesimal deformations
For a problem involving infinitesimal deformation (where shape changes are characterized by the infinitesimal strain tensor and rotation tensor) all the stress measures defined in the preceding section are approximately equal.
To
see this, write the deformation gradient as
The same procedure will show that material
stress and Cauchy stress are approximately equal, to within a term of order
2.2.6 Principal Stresses and directions
For any stress measure, the
principal stresses
Clearly,
The term `left’ eigenvector and eigenvalue indicates that the vector multiplies the tensor on the left. We will see later that Cauchy stress and material stress are both symmetric. For a symmetric tensor the left and right eigenvalues and vectors are the same.
Note that the eigenvectors of a symmetric tensor are orthogonal. Consequently, the principal Cauchy or material stresses can be visualized as tractions acting normal to the faces of a cube. The principal directions specify the orientation of this special cube.
One can also show that if
In
the same vein, the largest shear stress can be shown to act on the plane with
unit normal vector
2.2.7 Hydrostatic and Deviatoric Stress; von Mises effective stress
Given the Cauchy stress
tensor
The
Hydrostatic stress is defined as The
Deviatoric stress tensor is defined as The
Von-Mises effective stress is
defined as The hydrostatic stress is a measure of the pressure exerted by a state of stress. Pressure acts so as to change the volume of a material element.
The deviatoric stress is a measure of the shearing exerted by a state of stress. Shear stress tends to distort a solid, without changing its volume.
The
Von-Mises effective stress can be
regarded as a uniaxial equivalent of a multi-axial stress state. It is used in many failure or yield
criteria. Thus, if a material is known
to fail in a uniaxial tensile test (with
The hydrostatic stress and von Mises stress can also be expressed in terms of principal stresses as
The hydrostatic and von
Mises stresses are invariants of
the stress tensor
2.2.8 Stresses near an external
surface or edge
Note that at an external surface at which tractions are prescribed, some components of stress are known. Specifically, let n denote a unit vector normal to the surface, and let t denote the traction (force per unit area) acting on the surface. Then the Cauchy stress at the surface must satisfy
For
example, suppose that a surface with normal in the
The
stress state at an edge is even simpler.Â
Suppose that surfaces with normals in the
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(c) A.F.
Bower, 2008 |