Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

50 1 Introduction to Vectors and Tensors

Note that, especially in the mathematical community, the alternative definition divA =

of the divergence of a second-order tensor field A is frequently used.

The gradient (or derivative) of a smooth (second-order) tensor field A(x) is defined to be a (third-order) tensor.field. It is denoted by gradA (or VA, or DA/Dx, or

more explicitly by V ®A). Using ( 1.264)3 and representation ( 1.59), we may write

Laplacian and Hessian. The vector operator V dotted into itself gives the Laplacian (or the Laplacian operator) \72 (or V · V, or in the literature sometimes denoted by ~). With the rules ( 1.261 )1, ( 1.264)1 and properties ( 1.2 I), ( l.22h we find that

The Laplacian V2 operated upon a scalar field cI> yields another scalar field. One example is Laplace's equation \72 cI> = 0. Another example is the inhomogeneous Laplace's equation often referred to as Poisson's equation, i.e. \72 <I> = w. The differential equations of Laplace and Poisson occur in many fields of physics and engineering, for example, in electrostatics, in elasticity theory, in heat conduction problems of solids and so on.

The operator VV (or more explicitly denoted by V ® V) is called the Hessian.

With the rules ( 1.261 )i and ( 1.264h we obtain

\7 2 u = o (see eq. ( 1.285)). For this case the vector field u(x) is said to be harmonic. If a scalar field cI> satisfies Laplace's equation \72 cI> = 0, then cI> is said to be harmonic.

EXERCISES

1. Consider a smooth scalar field <I> = x 1:z:2:r3 - :r: 1• Find a vector n of unit length normal to the level surface cl>= canst passing through {2, 0, 3).

coordinates (0, 1), for increasing values of:1: 1•

4.If U = u(x) == :r1:1:2J:ae1 + :r1:r2C2 + :r1 e3, determine divu, curio, graclu, V72u, respectively. Verify that (1.285) is satisfied.

5.Apply the operator V to products of smooth scalar fields cl>, \JI, vector fields u, v and tensor fields A. Establish the important identities

6. Consider the inverse square law defined by the particular vector field

x

u =a lxFI '

with the positive constant a and x = :ciei. The region of u is the three-dimensio- nal space, excluding the origin <lxl # 0). Note that in fluid mechanics this relation characterizes the velocity field of a point source.

(a) Show that the vector field u(x) is harmonic.

(b} Derive a scalar field cl> whose gradient is u. (<I> == -a/lxl>

and let n be the outward unit normal field acting along the surface s. Show that

2.Let u and A be smooth vector and tensor fields defined in v and on .s and let n be the outward normal to s. Use index notation to show that

v

where e is the third-order permutation tensor as expressed in eq. ( 1.143 ).

3.Let cI> and u be smooth scalar and vector fields defined on s and c and let n act on s. Show that

2 Kinematics

classes of materials, regard.less of their internal physical structure. In -order to explain the macroscopic behavior of physical objects, :first of all we must understand the motion and deformation that cause stresses in a material (or are caused by stress.es) arising from forces and moments. Thus, to study the motion and (finite) d.eformation of a continuum, i.e. kinematics, is essential and mainly the aim -of the following chapter.

The reader who requires additional information on the subject of this -chapte.r i-s referred to the monographs by, for example, MALVERN [ l 969], WANG and TRUESDELL

.(1973], CHADWICK [1976], TRUESDELL [1977], GURTIN [1981.a], CIARLBT :[1988],

TRUESDELL and NOLL [1992], MARSDEN and HUGHES [.1994) or OGDEN [1997].

2.1Configurations, and Motions of Continuum Bodies

Under an electron microscope we see the discontinuous atomic structure of matter. The molecules may be crystalline or .randomly oriented. Between the particles there are large gaps. Theories considering the discrete structure of matter are ·molecular and atomistic theories. They are based on a discrete particle approach. For macroscopic systems such theories tend .to become too complicated to yield the desired results and therefore do not.meet our needs. However, under certain circumstances the microscopic approach is indispensable for the study of physical phenomena. A review of atomistic models is :presented in the work of ORTIZ f1999, and references the.rein_].

Notation of a particle nod a continuum body. Macroscopic systems usually can be described successfully with a continuum a_pproach (macroscopic .approach). Such an approach leads to the continuum theory. The continuum theory has been developed .independently of the molecular and atomistic theory and is meeting our needs. A fundamental assu-mp.tion therein states that a body, denoted by B, may be viewed as having a continuous (or at least a piecewise continuous) distribution of matte.r .in space and time. The body is imagined as being a composition of a (continuous) set of parti.. cles (or continuum particles or material points), represented by P E B as indicated

lt is important to note that the notion 'particle' (or 'continuum .particle' or 'material point') refers to a part of a body and does not imply any assoc.iation with the point mass -of Newtonian mechanics or the discrete particle of the atomistic theory as .mentioned above. A typical continuum particle is an accumulation of a large number of molecules, yet is small enough to be considered as a particle. The behavior of a continuum partic.le is a consequence of the collective behavior of all the molecules constituting that particle.

Hence, in a macroscopic study we are concerned with the mec.hanics of a body in which both mass and volume are continuous (o.r at least piecewise continuous) .func-

58 2 Kinematics

time to another it occupies a continuous sequence of geometrical regions denoted by f201 ••• , n. Hence, every particle P of B corresponds to a so-cal led geometrical point having a position in regions n01 ••. , n. The .regions which are occupied by the continuum body Bat a given time-tare known as the configurations of Bat that time t. The continuum body B may have .infinitely many configurations in space.

The geometrical regions are determined uniquely at any instant of ti.me. Region n0 with the position .of a typical point X corresponds to a fixed reference time. The region is referred to as the fixed -reference (or undeformed) configuration of the body

B. A region at ·inithd time t = 0 is referred to as the initial configuration. We agree subsequently that the .initial configuration coincides with the reference configuration; hence, the reference time is at t = 0. Note dynamics the initial configuration is often not chosen as the referenc-e configuration. Now, the po.int X has the position of a particle occupied by P E B at t = 0, and P may he identified by the position vector (or referential posUion) X of point X relative to the fixed origin 0.

We now assume that the region 0 0 of space moves to a new region

occupied by the continuum body B at a subsequent time t > 0. The configuration of Batt is its so-called current (or deformed) configuration. We relate a typical point

X of the reference con.figuration to a point x of the current configuration occupied by a particle P EB at times t = 0 and t > 0, respectively. The :position vector (or current pos"ition) x serves as .a label for the .associated point x with .respect to the ·fixed origin 0. It is often convenient to call X po.int X, associated with the particle P E ·B at t = 0, and to call x point x (or place x)., associated _with P E B at :t.

The components of vectors X = )(AEA and x ·=:cu.ea are considered as being along the axes introduced. We .Jabel .1X";t, il = 1, .2, ·3, as the material (or referential) coordi~ nates -of point X and :r:a, a = 1, 2, 3, as the spatial (or current) coordinates of point x. ·we have assumed that the origins of the sets {EA} and {Ca} of basis vectors -coincide. ln addition, we have agreed to have the same reference frame for the reference and current configurations, which is why we set the basis {EA} identical to {ea}. We use just {ea} in the following.

To denote scalar, vector and tensor-quantities we use uppercase letters when they are evaluated in the reference configuration, and lowercase letters for corresponding quantities in the current configuration. Sometimes we employ the index zero for quantities acting in the reference configuration (for example, n0 , p0 , a0 ... ). This convention is often used in textbooks of continuum mechanics; however, it occasionally leads to con- "fticts with the ·general aim of Chapter l where we agreed to use lowercase, bold-face letters for vectors and uppercase, bold-face letters for second-order tensors.

For example. we use X for the position vector of a point corresponding to the reference configuration despite the convention of Chapter 1 that uppercase, bold-face Latin letters usually denote second-order tensors. A dear and unique notation .is difficult to establish, considering that certain quantities in continuum mechanics are characterized