Introduction to Elasticity Part 10 pps

Figure 6: The 6-element truss in its original and deformed shape.General Stress Analysis The element stiffness matrix could be formed exactly for truss elements, but this is not the case

Figure 6: The 6-element truss in its original and deformed shape.

General Stress Analysis

The element stiffness matrix could be formed exactly for truss elements, but this is not the casefor general stress analysis situations The relation between nodal forces and displacements arenot known in advance for general two- or three-dimensional stress analysis problems, and anapproximate relation must be used in order to write out an element stiffness matrix

In the usual “displacement formulation” of the finite element method, the governing tions are combined so as to have only displacements appearing as unknowns; this can be done byusing the Hookean constitutive equations to replace the stresses in the equilibrium equations bythe strains, and then using the kinematic equations to replace the strains by the displacements.This gives

Of course, it is often impossible to solve these equations in closed form for the irregular ary conditions encountered in practical problems However, the equations are amenable todiscretization and solution by numerical techniques such as finite differences or finite elements.Finite element methods are one of several approximate numerical techniques available forthe solution of engineering boundary value problems Problems in the mechanics of materialsoften lead to equations of this type, and finite element methods have a number of advantages

bound-in handlbound-ing them The method is particularly well suited to problems with irregular geometriesand boundary conditions, and it can be implemented in general computer codes that can beused for many different problems

To obtain a numerical solution for the stress analysis problem, let us postulate a function

˜

Here the index j ranges over the element’s nodes, uj are the nodal displacements, and the Nj are

“interpolation functions.” These interpolation functions are usually simple polynomials erally linear, quadratic, or occasionally cubic polynomials) that are chosen to become unity atnode j and zero at the other element nodes The interpolation functions can be evaluated at anyposition within the element by means of standard subroutines, so the approximate displacement

(gen-at any position within the element can be obtained in terms of the nodal displacements directlyfrom Eqn 5

Figure 7: Interpolation in one dimension

The interpolation concept can be illustrated by asking how we might guess the value of afunction u(x) at an arbitrary point x located between two nodes at x = 0 and x = 1, assuming

we know somehow the nodal values u(0) and u(1) We might assume that as a reasonableapproximation u(x) simply varies linearly between these two values as shown in Fig 7, andwrite

u(x) ≈ ˜u(x) = u0(1− x) + u1(x)or

˜u(x) = u0N0(x) + u1N1(x),

(

N0(x) = (1 − x)

N1(x) = xHere the N0 and N1 are the linear interpolation functions for this one-dimensional approxima-tion Finite element codes have subroutines that extend this interpolation concept to two andthree dimensions

Approximations for the strain and stress follow directly from the displacements:

A “virtual work” argument can now be invoked to relate the nodal displacementuj appearing

at node j to the forces applied externally at node i: if a small, or “virtual,” displacement δui issuperimposed on node i, the increase in strain energy δU within an element connected to thatnode is given by:

Equation 12, with the integral replaced by numerical integrations of the form in Eqn 13, isthe finite element counterpart of Eqn 3, the differential governing equation The computer willcarry out the analysis by looping over each element, and within each element looping over theindividual integration points At each integration point the components of the element stiffnessmatrixkij are computed according to Eqn 12, and added into the appropriate positions of the

Kij global stiffness matrix as was done in the assembly step of matrix truss method described inthe previous section It can be appreciated that a good deal of computation is involved just informing the terms of the stiffness matrix, and that the finite element method could never havebeen developed without convenient and inexpensive access to a computer

Stresses around a circular hole

We have considered the problem of a uniaxially loaded plate containing a circular hole in previousmodules, including the theoretical Kirsch solution (Module 16) and experimental determinationsusing both photoelastic and moire methods (Module 17) This problem is of practical importance

—- for instance, we have noted the dangerous stress concentration that appears near rivet holes

— and it is also quite demanding in both theoretical and numerical analyses Since the stressesrise sharply near the hole, a finite element grid must be refined there in order to produceacceptable results

Figure 8: Mesh for circular-hole problem

Figure 8 shows a mesh of three-noded triangular elements developed by the felt-velvet

graphical FEA package that can be used to approximate the displacements and stresses around

a uniaxially loaded plate containing a circular hole Since both theoretical and experimentalresults for this stress field are available as mentioned above, the circular-hole problem is a goodone for becoming familiar with code operation

The user should take advantage of symmetry to reduce problem size whenever possible, and

in this case only one quadrant of the problem need be meshed The center of the hole is keptfixed, so the symmetry requires that nodes along the left edge be allowed to move verticallybut not horizontally Similarly, nodes along the lower edge are constrained vertically but leftfree to move horizontally Loads are applied to the nodes along the upper edge, with each loadbeing the resultant of the far-field stress acting along half of the element boundaries betweenthe given node and its neighbors (The far-field stress is taken as unity.) Portions of the feltinput dataset for this problem are:

free Tx=u Ty=u Tz=u Rx=u Ry=u Rz=u

slide_x color=red Tx=u Ty=c Tz=c Rx=u Ry=u Rz=u

slide_y color=red Tx=c Ty=u Tz=c Rx=u Ry=u Rz=u

end

The y-displacements and vertical stresses σy are contoured in Fig 9(a) and (b) respectively;these should be compared with the photoelastic and moire analyses given in Module 17, Figs 8and 10(a) The stress at the integration point closest to the x-axis at the hole is computed

to be σy,max = 3.26, 9% larger than the theoretical value of 3.00 In drawing the contours ofFig 9b, the postprocessor extrapolated the stresses to the nodes by fitting a least-squares planethrough the stresses at all four integration points within the element This produces an evenhigher value for the stress concentration factor, 3.593 The user must be aware that graphicalpostprocessors smooth results that are themselves only approximations, so numerical inaccuracy

is a real possibility Refining the mesh, especially near the region of highest stress gradient atthe hole meridian, would reduce this error

Figure 9: Vertical displacements (a) and stresses (b) as computed for the mesh of Fig 8

Problems

1 (a) – (h) Use FEA to determine the force in each element of the trusses drawn below

2 (a) – (c) Write out the global stiffness matrices for the trusses listed below, and solvefor the unknown forces and displacements For each element assume E = 30 Mpsi and

A = 0.1 in2

3 Obtain a plane-stress finite element solution for a cantilevered beam with a single load atthe free end Use arbitrarily chosen (but reasonable) dimensions and material properties.Plot the stresses σx and τxy as functions of y at an arbitrary station along the span; alsoplot the stresses given by the elementary theory of beam bending (c.f Module 13) andassess the magnitude of the numerical error

4 Repeat the previous problem, but with a symmetrically-loaded beam in three-point ing

bend-Prob 1

Prob 2

5 Use axisymmetric elements to obtain a finite element solution for the radial stress in athick-walled pressure vessel (using arbitrary geometry and material parameters) Comparethe results with the theoretical solution (c.f Prob 2 in Module 16)

Prob 3

Prob 4

ENGINEERING VISCOELASTICITY

David RoylanceDepartment of Materials Science and EngineeringMassachusetts Institute of Technology

Cambridge, MA 02139October 24, 2001

This document is intended to outline an important aspect of the mechanical response of polymersand polymer-matrix composites: the field of linear viscoelasticity The topics included here areaimed at providing an instructional introduction to this large and elegant subject, and shouldnot be taken as a thorough or comprehensive treatment The references appearing either asfootnotes to the text or listed separately at the end of the notes should be consulted for morethorough coverage

Viscoelastic response is often used as a probe in polymer science, since it is sensitive tothe material’s chemistry and microstructure The concepts and techniques presented here areimportant for this purpose, but the principal objective of this document is to demonstrate howlinear viscoelasticity can be incorporated into the general theory of mechanics of materials, sothat structures containing viscoelastic components can be designed and analyzed

While not all polymers are viscoelastic to any important practical extent, and even fewerare linearly viscoelastic1, this theory provides a usable engineering approximation for many

applications in polymer and composites engineering Even in instances requiring more elaboratetreatments, the linear viscoelastic theory is a useful starting point

When subjected to an applied stress, polymers may deform by either or both of two tally different atomistic mechanisms The lengths and angles of the chemical bonds connectingthe atoms may distort, moving the atoms to new positions of greater internal energy This is asmall motion and occurs very quickly, requiring only≈ 10−12 seconds.

fundamen-If the polymer has sufficient molecular mobility, larger-scale rearrangements of the atomsmay also be possible For instance, the relatively facile rotation around backbone carbon-carbon single bonds can produce large changes in the conformation of the molecule Depending

on the mobility, a polymer molecule can extend itself in the direction of the applied stress, whichdecreases its conformational entropy (the molecule is less “disordered”) Elastomers — rubber

— respond almost wholly by this entropic mechanism, with little distortion of their covalentbonds or change in their internal energy

1For an overview of nonlinear viscoelastic theory, see for instance W.N Findley et al., Creep and Relaxation

of Nonlinear Viscoelastic Materials, Dover Publications, New York, 1989.

The combined first and second laws of thermodynamics state how an increment of mechanicalwork f dx done on the system can produce an increase in the internal energy dU or a decrease

In contrast to the instantaneous nature of the energetically controlled elasticity, the formational or entropic changes are processes whose rates are sensitive to the local molecularmobility This mobility is influenced by a variety of physical and chemical factors, such as molec-ular architecture, temperature, or the presence of absorbed fluids which may swell the polymer.Often, a simple mental picture of “free volume” — roughly, the space available for molecularsegments to act cooperatively so as to carry out the motion or reaction in question — is useful

con-in con-intuitcon-ing these rates

These rates of conformational change can often be described with reasonable accuracy byArrhenius-type expressions of the form

rate∝ exp−E†

Figure 1: Temperature dependence of rate

Conversely, at temperatures much less than Tg, the rates are so slow as to be negligible.Here the chain uncoiling process is essentially “frozen out,” so the polymer is able to respondonly by bond stretching It now responds in a “glassy” manner, responding instantaneously

and reversibly but being incapable of being strained beyond a few percent before fracturing in

is used in the viscoelastic regime, as viscoelastic response can be a source of substantial energydissipation during impact

At temperatures well below Tg, when entropic motions are frozen and only elastic bond formations are possible, polymers exhibit a relatively high modulus, called the “glassy modulus”

de-Eg, which is on the order of 3 GPa (400 kpsi) As the temperature is increased through Tg, thestiffness drops dramatically, by perhaps two orders of magnitude, to a value called the “rubberymodulus” Er In elastomers that have been permanently crosslinked by sulphur vulcanization

or other means, the value of Er is determined primarily by the crosslink density; the kinetictheory of rubber elasticity gives the relation as

If the material is not crosslinked, the stiffness exhibits a short plateau due to the ability

of molecular entanglements to act as network junctions; at still higher temperatures the glements slip and the material becomes a viscous liquid Neither the glassy nor the rubberymodulus depends strongly on time, but in the vicinity of the transition near Tg time effects can

entan-be very important Clearly, a plot of modulus versus temperature, such as is shown in Fig 2, is avital tool in polymer materials science and engineering It provides a map of a vital engineeringproperty, and is also a fingerprint of the molecular motions available to the material

Figure 2: A generic modulus-temperature map for polymers

3 Phenomenological Aspects

Experimentally, one seeks to characterize materials by performing simple laboratory tests fromwhich information relevant to actual in-use conditions may be obtained In the case of vis-coelastic materials, mechanical characterization often consists of performing uniaxial tensiletests similar to those used for elastic solids, but modified so as to enable observation of thetime dependency of the material response Although many such “viscoelastic tensile tests” havebeen used, one most commonly encounters only three: creep, stress relaxation, and dynamic(sinusoidal) loading

Creep

The creep test consists of measuring the time dependent strain (t) = δ(t)/L0 resulting fromthe application of a steady uniaxial stress σ0 as illustrated in Fig 3 These three curves are thestrains measured at three different stress levels, each one twice the magnitude of the previousone

Figure 3: Creep strain at various constant stresses

Note in Fig 3 that when the stress is doubled, the resulting strain in doubled over its fullrange of time This occurs if the materials is linear in its response If the strain-stress relation

is linear, the strain resulting from a stress aσ, where a is a constant, is just the constant a timesthe strain resulting from σ alone Mathematically,

(aσ) = a(σ)This is just a case of “double the stress, double the strain.”

If the creep strains produced at a given time are plotted as the abscissa against the appliedstress as the ordinate, an “isochronous” stress-strain curve would be produced If the material

is linear, this “curve” will be a straight line, with a slope that increases as the chosen time isdecreased

For linear materials, the family of strain histories (t) obtained at various constant stressesmay be superimposed by normalizing them based on the applied stress The ratio of strain tostress is called the “compliance” C, and in the case of time-varying strain arising from a constantstress the ratio is the “creep compliance”:

Ccrp(t) = (t)

σ