Text 3. External and Internal Loads

A body undergoes deformation when subjected to external and internal forces. These forces may be mechanical, electrical, chemical, or of some other origin. Mechanical forces acting on a particle, according to Newtonian mechanics, are functions of the position vector x, velocity vector v and the time t. However, in continuum mechanics the motion of a collection of many particles is analyzed so that the forces may depend on the position and velocity of all particles in the collection at all past times. Since we do not identify particles, this means that the relative deformation of particles with respect to neighboring ones and the history of this defor­mation may come into play. Thus, forces may depend on various order spatial gradients, their various time rates and integrals, as well as electrical and chemical variables.

As in classical mechanics, the forces are not defined. To the undefined quantities such as position, time, and mass there are added two more, namely, the force F and the couple M acting on bodies. They are vectorial quantities given by

(3.2.1) ,

These quantities are known a priori. The total force 3F consists of the vector sum of all forces acting on the body, and the total couple M consists of two parts: the total moment

of the individual forces about a point (for example, origin O)and that of couples dM.

From a continuum point of view, whatever the origin may be, we divide the forces and couples into three categories.

 

Extrinsic Body Loads. These are the forces and couples that arise from the external effects. They act on the mass points of the body. A load density per unit mass is assumed to exist. The extrinsic body loads per unit volume are called volume or body loads. Examples are the force of gravity and electrostatic forces. Extrinsic body loads are not objective. The trans­formations of these loads are deduced from the basic Axioms 2 and 3 introduced in Section 2.9.

 

Extrinsic Surface Loads (Contact Loads). These loads arise from the action of one body on another through the bounding surface. The surface density of these loads is assumed to exist. The extrinsic surface force per unit area is called the surface traction, and the extrinsic surface couple per unit area is called the surface couple. Surface tractions and surface couples de­pend on the orientation of the surface on which they act. The hydrostatic pressure acting on the surface of a submerged body and surface tractions produced by an external electrostatic field are examples of extrinsic surface loads.

 

Internal Loads (Mutual Loads). These are the result of the mutual action of pairs of particles that are located in the interior of the body. According to Newton's third law, the mutual action of a pair of particles consists of two forces acting along the line connecting the particles, equal in magnitude, and opposite in direction to one another. Therefore the resultant internal force is zero. Mutual loads are objective.

Answer the questions.

A. When does a body undergo deformation?

B. How is the motion of a collection of many particles analyzed in continuum mechanics?

C. What may forces depend on?

D. What does the total force 3F consist of?

E. What categories can the forces and couples be divided into from a continuum point of view?

What are the key words in the text?

3. Give a short summary of the text using the key words.

 

Text 4. Objective Tensors

The physical properties of materials are not dependent on the coordinate frame selected. It is intuitively clear that whether the observer is at rest or in motion, the material properties he observes should be the same. If this viewpoint is accepted, then the measurements made in one frame of reference are sufficient to determine the material properties in all other frames that are in rigid motion with respect to one another.

In the formulation of physical laws, it is desirable to employ, as far as possible, quantities that are independent of the motion of the observer. Such quantities are called objective or material frame-indifferent. For exam­ple, the location of a point will appear different to observers located at different places. Similarly the velocity of a point is dependent on the velocity of the observer. Therefore, these quantities are not objective. On the other hand, the distance between two points and the angles between two directions are independent of the rigid motion of the frame of reference (the observer). Newton's laws of motion have long been known to be valid only in a special frame of reference called the galilean frame. A galilean frame differs from a fixed reference frame by a constant translatory velocity. Attempts to free the principles of mechanics from the motion of the observer were resolved by Einstein's theory of general relativity.

We wish to stay in the domain of classical mechanics with regard to basic axioms. However, we would like to employ the principle of objectivity in the description of material properties.

Let a rectangular frame F be in relative rigid motion with respect to another one, F´. A point with rectangular coordinates at time t in F will have the rectangular coordinates at time t' in F´. Since the frames are in rigid motion with respect to each other, we have

(2.10.1) t' = t – a

where a is a constant allowing us to select the origin of time different in x' than in x, and Q (t) and b (t) are functions of time alone, of which Q (t) is subject to

(2.10.2) Q_klQ_ml = Q_lkQ,_m =

These conditions are the usual conditions satisfied by the cosine directors of x' with respect to x. From (2.10.2) it follows that

(2.10.3) det Q = ±1

The rigid motions exclude the minus sign on the right-hand side, that is

(2.10.4) det Q =1