Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

The Concept of Stress

./.. I~.. the previous chapter smne kinematical aspects of the motion and deformation of ::·i:\:· i1 continuum body were discussed. Motion and deformation give rise to interactions

_(-.-_:.·between the material and neighboring material in the inte.rior part of the body. One /}consequence of these interactions is stress, which has physical dimension force per unit

//:._-:_·o.f area. The notion of stress, which .is responsible for the deformation of materials, .is

}._:\::~·rucial in continuum mechanics.

we consider a deformable body during a finite motion. For that body concept of stress and discuss the properties of traction vectors and

::_\,:::>stress tensors in different descriptions. We continue with a brief review of the de~ :~>i;:ter;inination of extremal stress values, noting that this topic has already been studied

:·i:-.:::-:\!.xtensively in numerous other boo~ks on mechanics.

F/ , Since the current configuration of many problems, especially those involving solids,

·:..:/i~..not known_, it is therefore not convenient to work with stress tensors which are ex-

( p'tessed in terms of spatial coordinates. For some cases it is more convenient to work

i"_::_::·._.: \,r.ith stress tensors that are referred to the reference configuration or an intermediate

: ....·.... ·.

(:::·::.-:~.~-n_figuration. Some important forms of alternative stress measures, which are associ-

)}..·ated with the names of Pio/a, Kirchhoff, Biol~ Green_, Nagluli or Mandel, are provided..

·.·.·.

\:._:._:"·._":-__.·: ...

:f{3.l. Traction Vectors, and Stress Tensors

[H~re we introduce surface tractions which are related to either the current or the refer-

//_:._-·-~-~·c·e configuration. They may be expressed in terms of unique stress fields acting .on a

/:.:{.._ribimal vector to a plane surface. Stress components such as normal and shear stress-es

{://·~[~:·introduced.

):::_:·:_·..;:_::::-: ..

.:=..({:.Surface tractions. We focus .attention on a deformable continuum body B -occu-

"i,/pying an arbitrary region n of physical space with boundary surface an at time t, as

'(shown in Figure 3.1.

::·.··:.·

109

i· The passage from u to P and back is known as the Piola trans/ormation (compare with _:·the definition on p. 83). Strictly speaking, in order to obtain the two-point tensor P,

::/..with components PaA, we have performed a Piola transformation on the second index

·... ·

:::::..of tensor .u, with components <1ab·

.. For convenience, we omit subsequently the arguments of .the tensor quantities. The ::-.:explicit expression for the symmetric Cauchy stress tensor results as the inverse of

.·:relation (3.8), i.e.

...

>.·::_:·:_-.:.: .. Conse-quently, the second-order tensor Pis, in general, not symmetric and has nine

:::::Independent com_ponents PaA·

;.·:·:·.·····

:::::}'he Cauchy stress tensor for a certain point of a body is given by its matrix represen- ·/fation as

:_:}Jetermine the Cauchy traction vector t and the first Piola-Kirchhoff traction vector T

ll4 3 The Concept of Stress

while detF = J = 1. The components of the .first Piola-Kirchhoff stress tensor read, according to (3.8), as

In order to find the outward unit normal N in the reference configuration we recall Nanson's formula NdS = J-1FTnds. Hence, with the transpose of matrix [F], J = 1 and knowing that n = e2 we find that

rection. The magnitude of Tis twice that o.f t, because_, in view of (3.15), the deformed

area is half the undeformed area. II

...... ,............... ,...... ,.. :··· ................ :.. :·'''"'"··· ...................... •:··•.""· ......................... ······.···········t·:·'·:·····.····."·t."•t·:·····:····.-. ..·.:····:·:··.·······t.··t·····:·········:··.··:··············.·-.··,··:···········.···:·.···················.···················:··:·········.······:···:··········

In view of Cauchy's stress theorem (3.3) 1, te,, = ueb., b = 1, 2, 3, characterize the three Cauchy traction vectors acting on surface elements whose outward norma.ls point in respectively. We use the notation tc0 = t(x, t, ea) to .indicate

explicitly the dependence of the traction vector t on the basis vectors e11 •

In matrix notation, the columns of [uJ can be .identified as the components of traction vectors acting on planes perpendicular to e1, ·e2 , e3, respectively.

(see the three faces of a cube illustrated in Figure 3.3), characterizing the .state of stress

at a certain point.

obtain

(3.22)

Note that the quantity a 21 is the component of te1 in the direction of e2 (compare with Figure 3.3). •

·-··~·-.........~------=----"..-<(•--.-~..·~.-.:Y-••""C":"--- ...- ..• J,~_,,_••'~'-•,.....,__;-..-- ,· - .-- .-·•-=---(V..~•_..:..,•"'~--'""'':-·---:"-"··-,·,--~-~~

In .order to introduce a sign convention for the stress components look at Figure 3.3. We denote those faces of a cube seen by the observer as positive. The so-called negative faces of the cube are not visible to the observer. When the stress components aab

are positive scalars, then the components of the traction vectors are defined to be

.positive, as illustrated in Figure 3.3. A negative vector component on positive faces of the cube points .in the ·opposite direction. Vector components on a negative face are just the opposites of those ·on the associated positive face.

Let the Cauchy traction vector tn ·= t{x, t, n) for a given current position x E fl at time t act on an arbitrary oriented surface element The surface is characterized by an outward unit norm.al vector n and a unit vector m embedded in the surface satisfying the property m · n = 0..

__,..,.... \_tn = tU + tk

.,....

,,

.......

...

/tU=crn

n

Figure 3..4 Normal and shear stresses at current position x..

According to Figure .3.4, t0 may be resolved into the sum of .a vector along the_ normal n to the plane, denoted by t!L and a vector perpendicular to n, denoted by t;~:·. With the rule ( 1.53) we find that

(323)

(compare with the equivalent equations (1J25) and (.1.126)), where P.U and P* are projection tensors of order two defined by

P~=n0n, r; ·=m ·0 m = I - n ® n , (3.25)

with properties according to ( 1.127)-(1.130).

The lengths of :tH and t; are cal led the normal stress .a and shear stress (or tangen..

. .

·tial stress) T acting on the surface element cons-idered, respectively. Using eq. (3.3).i

the explicit expressions for a and rare

\_::(compressive stresses are characterized as positive and the tensile stresses as negative.

\:/.!.t is important to note that a tensile stress is a fundamental different type of loading -~://than a cmnpression stress. However, the sign of a shear stress has no intrinsic physical

).::'.relevance, the type of loading is the same.

}/·•.· From Figure 3.4 we find the useful geometrical relation

mx> Consider the schematic diagram of Figure 3.3. Each component D"a11 (a = 1, 2, 3; i.{\#o summation), i.e. cr1L, a22 , cr:1:.h is a norma] stress acting in the direction of the unit

;:\{{;._Vector .normal to a surface element. These stresses are the diagonal elements of the

!/((~.t.~ess matrix [u] gjven in eq. (3.6h. Each remaining c·ompone.nt a 0 b (a # b)~ i..e.

·/):·:_d·1~, a1a, ..., is a shear stress acting tangentially to a surface ele:ment. Shear stresses

.i:ff~re the off-diagonal elements of the stress matrix [u].

:._:.:: ~-:::- ": .·.

'{::::(:·:})..·_::· ..

t&i/u:l. Consider Cauchy traction vecto::t:~::::on two infinitesimal surface elements

{!f;\;}i·r.:=:·:·.··.:.at x with normals n and ii. Show that

2.At a certain point of a deformable body the Cauchy stress components a.re given with respect to a ·:i: 1, :c2 _, :1:a-coordi.nate .systemlas

14 _')~

[u]= 4 0 . kN/m2

[ -2 0 3

(a)Find the components of the Cauchy traction vector tn and the length of the

vector along the normal to the plane that passes throu:gh this point and that is parallel to the p.lane 2-:r 1 + 3:i:2 + X:J = 5.

(b)Find the length of tn and the angle that t11 makes with the normal to the. p.lane.

(c) Find the stress components .aa.b along a new orthonormal basis {ea}, a =

1, 2, 3, given by the following transformation

(compare with Section 1.5), where {Ca} characterizes the old orthononl)a~ basis.

3. Consider the Cauchy stress matrix in the form

and find the components of the Cauchy traction vector t0 at the· point with coor- ·. dinates (1/2, V3/2, -1) on the surface :1:i + :z:§ + ~1::~ = 0.

4. At a -certain point of a body the components of the Cauchy stress tensor are given

by

{a) Find the components -of the Cauchy traction vector tn at the point on the_/ plane whose normal has direction ratios .3 : 1 : - 2.

(b) Find the normal and shear components of t on that plane.