EN222: Mechanics of Solids

  

 

 

 

 

 

 

 

 

 

 

 

Division of Engineering

    Brown University

 

 

1.2 Kinetics

 

We proceed to review measures of internal and external forces acting on solids.

 

Applied Forces

 

 

A solid body can be loaded in two ways. 

(i) A force can be applied to its boundary: examples include fluid pressure, wind loading, or forces arising from contact with another solid. 

(ii) The solid can be subjected to body forces, which act on the interior of the solid.  Examples include gravitational loading, or electromagnetic forces.

 

 The surface traction vector t at a point on the surface represents the force acting on the surface per unit area of the deformed solid.

 

 

Formally, let dA be an element of area on a surface.  Suppose that dA is subjected to a force .  Then

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 body force vector denotes the external force acting on the interior of a solid, per unit mass.

Formally, let dV denote an infinitesimal volume element within the deformed solid, and let  denote the mass density (mass per unit deformed volume).  Suppose that the element is subjected to a force .  Then

 

The resultant body force acting on any volume V within the deformed solid is

 

 Internal forces induced by external loading

     

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.

 

 The internal traction vector T(n) represents the force per unit area acting on a plane with normal vector n inside the deformed solid.

 

Formally, let dA be an element of area in the interior of the solid, with normal n.  Suppose that dA is subjected to a force .  Then

 

 

The internal tractions that act on material planes passing through a point are not independent.  Specifically, we may show that given the tractions acting on three mutually perpendicular planes, with normals parallel to , the traction on a plane with normal n follows as

This result is derived by considering forces acting on the `Cauchy Tetrahedron’ illustrated above, and applying Newton’s second and third laws. (see the EN175 notes to review how)

 

The result just quoted leads to the following definition:

 

 

 The Cauchy (true) stress tensor  is defined as

 

Note that some authors (Gurtin is a notable example) define Cauchy stress as the transpose of this definition, i.e.  . 

 

The components of Cauchy stress in a given basis can be visualized as the tractions acting on planes with normals parallel to each basis vector, i.e.

 

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.

 

Although Cauchy stress is easily interpreted physically, it is generally not the most convenient measure of internal force for defining constitutive relations, because it is not work-conjugate to any convenient measure of deformation. 

 

Note that we don’t need to know a reference configuration for the solid to define Cauchy stress.  This is not the case for other stress measures.  Consequently, we now introduce a deformation mapping and define basic deformation measures

 

 

Define       

Note               

 

  Kirchhoff stress

 

This stress measure has no obvious physical significance, but is work-conjugate to the stretch rate (i.e.  gives the rate of work done by internal stresses per unit reference volume)

 

 Nominal stress, or First Piola-Kirchhoff stress

Note again that some authors define nominal stress as the transpose of this measure.  The distinction is crucial here because the nominal stress tensor is not symmetric.

 

Nominal stress is work conjugate to the rate of change of deformation gradient, i.e.  is the stress power per unit reference volume.

 

Nominal stress does have a physical interpretation.  With reference to the figure above, let  be an infinitesimal element of area within the reference configuration, normal N.  When the solid is deformed, the area of this element changes to , and the normal to the deformed area element is n.

 

Let T(n) denote the traction acting on the deformed area element.  By definition, the force acting on the deformed area element is

 

The nominal stress tensor evidently satisfies

and so may be thought of as a measure of force per unit area of reference solid.

 

Material stress, or Second Piola-Kirchhoff stress

Nominal stress is work conjugate to the rate of change of Lagrange strain, i.e.  is the stress power per unit reference volume.

 

Material stress does not have a convenient physical interpretation – it is best thought of as a generalized force conjugate to the appropriate deformation measure (Lagrange strain).

 

Stresses for infinitesimal deformations

 

For infinitesimal motions we may linearize the stress measures defined above.  To this end, let

 with

define a deformation mapping.

 

Then

 

Hence

 

So  in problems involving small strains and rotations. 

 

 

 

Principal Stresses

 

Let  denote a Cauchy stress tensor.  Since  is symmetric it may be expressed as

where  are three mutually perpendicular unit vectors, and ,  and  are the three principal stresses (normally ordered so that  ).

 

Recall that ,  and  can be found by computing the eigenvalues of , while the principal directions follow as the normalized eigenvectors of .

 

Physically, this statement tells us that any stress state can be visualized as tractions acting normal to the faces of a cube.  The stress state is specified by the orientation of the cube and by the tractions acting normal to its faces.

 

One can also show that if , then  is the largest normal traction acting on any plane passing through the point of interest, while  is the lowest.  This is helpful in defining damage criteria for brittle materials, which fail when the stress acting normal to a material plane reaches a critical magnitude.

 

 

In the same vein, the largest shear stress can be shown to act on the plane at 45 degrees to the  and  axes, and its magnitude is .  This observation is useful for defining yield criteria for metal polycrystals, which begin to deform plastically when the shear stress acting on a material plane reaches a critical value.

 

One can evidently define principal values of material stress in the same way.  The issue of existence of principal values of nominal stress and their physical significance is left as an exercise.

 

Stress Invariants

 

When defining constitutive relations for isotropic solids (which must be independent of the basis used to define the stress or strain components) it is useful to use stress measures that are independent of the choice of basis.

 

Of course, the principal stresses provide one such choice.  However, they cannot be expressed in terms of stress components in a simple way.  More convenient, and commonly used stress invariants are

 

Hydrostatic stress

 

Von-Mises equivalent stress

 

 

Third invariant of deviatoric stress (no special name – its not used much – name it after yourself if you like)

 

where  is known as the deviatoric stress.

 

It is possible to define other stress invariants, of course, (and we shall do so quite freely in developing plasticity theories) but they can always be expressed in terms of the three invariants listed here.

Stress rates

 

You may recall that constitutive relations for plastic and viscoelastic solids are generally written as relations between strain rate and stress and stress rate.  For example, for small strain problems we relate stress rate to the elastic strain rate as

For finite strain problems we might expect to obtain a relationship between stretch rate D and the rate of Kirchhoff stress (recall that Kirchhoff stress is work conjugate to D and so is the stress measure of choice).  This is in fact the case, but the stress rate that appears in the expression is no simply the time derivative of Kirchhoff stress.

To understand why this is the case, visualize a rigid bar which rotates with constant angular velocity about the  axis.  Suppose the bar is loaded parallel to its axis inducing a stress .  It is easy to see that the material time derivative  is nonzero, because the principal axes of stress are rotating.  The stretch rate D vanishes, however.  So while our deformation measure correctly indicates that the bar is not changing its shape, our stress rate measure does not recognize that the material is actually experiencing a steady stress.

 

To correct this problem we define a stress rate that quantifies the rate of change of stress with respect to a frame that rotates with the material.  To this end, let  be a fixed basis, and let  be a basis that rotates with the bar.  Assume that at the instant shown the two sets of basis vectors coincide.

 

The stress state may be expressed in either basis as

 

The material stress derivative may be expressed as

 

The co-rotational stress rate is defined through the rate of change of stress components in a basis that rotates with the material, as

This stress rate does vanish for rotational motions.

 

To find the relationship between corotational stress rate and the rate of change of stress, we can differentiate the stress

Recalling that

where

is the spin tensor associated with the rigid rotation, we see that

 

Other stress rates are sometimes defined in the same spirit.

 

As a general rule, one does not need to worry about what stress rate to use in a constitutive law as long as the law is constructed based on sensible physics.  The appropriate stress rate then emerges naturally.  The discussion here is intended to explain the physical significance of the somewhat strange results that emerge for finite strain problems.

 

 

Balance Laws

 

The distribution of internal traction (or stress) within a solid must satisfy F=ma for any arbitrary volume of material within the solid.  We have already seen that this requires the existence of the Cauchy stress tensor. Linear and angular momentum balance impose further constraints on the distribution of stress within a solid.

Let  denote the Cauchy stress distribution within a deformed solid.  Let  denote the mass density (mass per unit deformed volume) at point x in the deformed solid. Assume that the solid is subjected to a body force , and let  and  denote the displacement, velocity and acceleration of a material particle at position   in the deformed solid.

 

 Linear momentum balance requires that

 

This result is derived by writing linear moment balance for an arbitrary internal volume V within the solid, then applying the divergence theorem.  To review the derivation, see the EN175 notes.

 

In terms of nominal and material stress

 

 

 

 

 Angular momentum balance requires that

or for other stress measures

 

The proof can be found in the EN175 notes.