Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

(by unequivoca] symbols.

· ··.. Whenever index notation is employed, as for the components of vectors and tensors, ·:::c-~nventionally we use uppercase letters for the reference configuration (for example, -:_'.·_'."·t.b BA ... ) and lowercase letters for the current configuration (for example, ~1:a., ba .. .).

:Motion. We .assume that the map X = ~0( P, ."t) . is a one-to-one correspondence :·between a particle P E B and the point X E Ho that 8 -occupies at the given instant of ·_:time t == 0 (see Figure 2.1). Furthermore, let the map"' act on B to produce the region

·..nat time t. The place x = ~(P_, t) that the particle P (evidently identified with X and !t). occupies at tis described by (in symbolic and index notation)

{?r. all X E f!0 and for all times t .. In (2.1) x is a vector field that specifies the place x of :.xfor all fixed t, .and -is called the motion of the body 8. The motion x carries points X 1·ocated at f20 to places x in the current configuration n. ·we assume subsequently that xpossesses continuous derivatives w..ith respect to space and time..

·.The parametric equation (2. I) determines successive positions x of.a typical particle

space. All successive points together form a curve in the Euclidean space which

\ve call the path line (or trajectory) of the particle P.

:>_·... ·. The motion xis assumed to be uniquely invertible. Consider (x1 t), the position of ·.point X, which is associated with the place x at time ·t, is specified uniquely by (2..I) as

.......... ; ... X = x·~1 (x,t) , (22)

\vith the inverse -motion,denoted by x-l., i.e. the .inverse of the mapping x~ as defined

-~~... ~q. (2.1). For a given time t, the inverse motion (2.2) carries points located at .n to

·.p_o_i_nts in the reference configuration n0 • .In (2.1) and (2..2) respectively the pairs

.... . . .

denote independent variables..

_:· ...~ A motion x of a body will generally change its shape~ position and orientation. to change its shape is said to be deformable. By a ~~formation x (or inverse deformation x- 1) of a body we mean a motion (or inverse

:motion) of a body that is independent of time.

:~,....... ,.,_...,.................... ·································:·:·:······.... .......,......................................"·······:·····:·.·.····························:·:·..••••'··'·····.··:'•.·:···'·."."·:·.'."•."•"..·····."•.'."·."·.··.··,..·····,:•.·,···. .: .•."•:.··

Let a deformed configuration of an initially .rectangular region with lengths L 1 and L2 be given as shown in Figure 2..2. The time-dependent angle fJ is

O(t) where w denotes the angular velocity.

:_'.="·:·:.-:Determine the motion of a .particle given by the position vector XE .n0 and time t In: particular, determine the -motion x = x(X, .t) at (L 1/2, .£2 /2, 1) and at time t = "/4, ·with w = 1~

: t = 0

. f!o

Figure 2.2 Deformed configuration of an initially rectangular region.

Solution. The reference configuration of the region 0 0 of interest is characterized by (Xt, ""\2 , ..X3_) .at t = 0, with 0 < .X1 < L1 , 0 < .X2 < L2 and 0 < .1Ya < 1.

The motion results in accord with (2.1)

Material .and spatial descri.ptions.. The so-called material (or referential) de-

scription is a characterization o.f the motion (or any other quantity) with respect to the material coordinates {.X1 , ~~2, ~Y3) and time t, given by eq. (2.1). In the material de- scription attention is. paid to a particle, and we observe what happens to the particle as it moves. Traditionally the material description .is often referred to as the Lagrangian

description (or :Lagrangian form). Note .that at t =0 we have the consistency condition X = x., .x·.:, = Xa.

The so-called Eule-rian (or spatial) description (or Eulerian form)~ is .a charac...

terization of -the -motion (or any other quantity) with .respect to the spatial coordinates (:t.t, x2 , x 3 ) and time t, given by eq. (2.2). In the spatial description attention is paid to a point in space, and we study what ·happens at the point as time changes.

In :fluid mechanics we quite often work in the Eulerian description in which we refer all relevant quantities to the position in space at time t. It is not useful to refer the quantities to the material coordinates ..iY1i, A = 1, 2, 3, at t = 0, which are, in general, not known -in fluid mechanics. However, in solid mechanics we use both types

of description. Due to the fact that the constitutive behavior of solids is often given in terms of material coordinates we often prefer the Lagrangian description.

EXERCISE

:I. Consider the two motions defined by the following sets of ·equations: x = o:t:ie + X ,

:1:2 = -fJ - 1e-111"cos:(8 +wt)

.~,

where .n, /3, ·/, c5 and w are scalar constants and e denotes a fixed unit vector. Show that the path lines are straight lines and circles, respectively.

2.2 Displacement, Velocity, A-cceleration Fields

Displacement field. The vector field

represents the displacement fie=ld of a typical particle and relates its position X in the undeformed configuration to its position x in the de.formed configuration at time t (see Figure 2.3). Relation (2.4) holds for all particles. The displacement fie.Id U is a function of the referential position X and ti.me t, which characterizes the material description (Lagrangian form) of the displacement field.

The displacement ·fieldin the spatial description {Eulerian form), denoted u, .is a

Representation (2.5) specifies the current position x of a particle at time t which results from its referential position X(x, t) plus its displacement u(x, t) from that position. Representation (2.4) -expresses the displacement of a particle at time t in terms of its ·referential position X. The two descriptions are related by means of the motion x = x(X, t), name-Iy

Quite often we need a formuJation in terms of the spatial coordinates (:1:J, :l:2, :i::i) chm·acterizing a fixed location in spac-e at time t. As mentioned above, such a description is known as the Eulerian or spatial description in which now we like to define the velocity field and the acceleration field.

:.":".... . The two -descriptions are transformed into -each other by using the motion X· In

The velocity components .and the accele·-ration components for V and A are de- ·:··;· noted by V4, AA, while for v and a we write Va, <La, respectively.

:-: : EXAMPLE 2.2 A certain motion of a continuum body in the material description is

: ·.:·_:· given in the form

:_.·:_.:Hon components in tenns of the material coordinates .XA •.A = l_, 2, 3, and time t foHow

:::·._:._from eqs. (.2.7) and have the forms

'.:../.:::·..:·~e_rived relations (2.l l) and (2..12). Thus, according to eqs. (2~.8) and (2.9) we find that

The subscript X, (.-YA), indicates the variable which is held constant while taking the partial differentiation of :F. In the following the subscripts are dropped to reduce the cumbersome notation,. However, in cases where additional information is needed, they will be employed.

The.material time derivative represents the rate at which the material field :F changes with time, seen by an observer following the path line of a particle.

Determine the material description of the velocity field V and the acce.leration field A of a particle at time .t as the time rate of change of the displacement of that particle.

Solution. By virtue of eqs. (2.1)2 and (2.4) we may derive from (2.7)

For some studies following in this text we introduce the useful relationship between the .material time derivative and the directional derivative. Considering F = F(X, 't),

which ·.can be shown by applying the definition of the directional derivative, i.e. (.l .266).

Material gradient .-of a material field. The material _gradient of a smooth material field F(X, t), denoted by Gra.dF(X, t) (with uppercase G), is obtained by deriving :F relative to the referential position X, (.XA), at a fixed time t, i.e.

Subsequently, the material dive·rgence and the material curl of :F are denoted by

DivF and CurLr (with uppercase D and C), respectively. They are divergence and

By analogy with the above, the ·material tim-e derivative of a smooth spatial field

where v = v{x, -tL (vu, a = 1, 2, 3), and its material ti.me .derivative v = v(x, :I:), (t}ll., a = 1, 2, 3), may also be viewed as the spatial description of a velocity and acceleration field, referred to brie-fly as the spatial velocity .field nnd the spatial acceleration .field,

(the arguments of the functions have been omitted for simplicity).

The first term on the right-hand side, name.1y avIat, describes the ·local accelera- tion (local rate of change of the velocity field). The second term is quadratically non- linear in the velocity field and describes the convective acceleration field (convective rate of change of the velocity field). Relation (2.27) turns out to be very useful, because the spatial acceleration field a(x, t) m~y be determined from v(x, t) without knowing the motion explicitly, which is an important question in applied :fluid mechanics,

By virtue of (2.7) 1 and (2.8), the spatial velocity field v(x, t) may be expresseµ as the material time derivative of the motion x = x(X, t), i.e.

and from (2.5) we find that v == ii. The spatial acceleration field re.ads by analogy with relation (2 .19) as

A very useful short-hand notation of lhe ope.rator D(•) /Dt which acts on spatial fields reads, in symbolic and index notation,

which should be compared with eqs. (.2.25) and (2.26). Relation (2.30) shows clearly the relationship between the ·material and spatial t\me derivatives.

68 2 Kinematics

--~---=--~---,~~""""":'''~-,,~,~-~,-~-.-~ .....,,-.,:·---·--·--- ___, -·--------------·-·-"-·'"·~,,.,.~~:,,, .._,.,.,,:·~~~:~~-,.~ .....,,,.~,,,,,, ~,,,,,,,,,~.,,.,,~ .....,,,,~,,,,. ,_.,,,.,,,~~-, .......,. ,~,.,,_,~.,,.~ ...-.,"':"""""-.......- ,.,_,.,.___,~---,·-~·-.-- ~-· .. -·-.-:·

EXAMPLE 2.4 In the current configuration of a continuum body a certain physical quantity is given in space and time according to

characterized by three variabJes corresponding to the coordinates and one to the time1 excluding the origin, i.e. lxl -# 0, t =F 0. Suppose that an observer moves with velocity

(2.32)

Find the time rate of change of <I> as seen by the observer.

Solution. In order to compute the material time derivative of the spatial fieid <I> apply relation (2.25)2. Therefore, we need to compute the spatial time derivative and the spatial gradient of <I>. The spatial time derivative of <I> gives

Note that the spatial vector field gTadcI> is harmonic and describes a point source at .a certain time t (compare with Exercise ·6 on p. 51 ).

By virtue of the derived solutions (2.33), (2.34) and the given velocity field (2.32) we obtain finally

(2.35)

which gives the desired solution {2.3 .l ), the material time derivative of <I> at x and at

and v = v(x), the associated motion is said to be steady and (2.27) reduces to a = (gradv)v. On the other hand, if the velocity field at each instant :is independent of position x, i.e. gradv = 0 and v = v{t), the associated motion is said to be uniform and (2.27) reduces to a If the velocity .field has components of the form