Computational Mechanics (1992) 10,369379
Computational
Mechanics
f; SpringerVerlag I ~~2
An adaptivemesh efinement echnique or the analysis of shearbandsin planestrain compression f a thermoviscoplastic olid
R. C. Batra and K.I. Ko
Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of MissouriMO 654010249, USA
Rolla, Rolla,
Abstract. We have developed an adaptive mesh refinement technique that generates elements such that the integral of thesecond nvariant of the deviatoric strainrate tensor over an element is nearly the same or all elements in the mesh. t is shownthat the finite element meshes so generated are effective n resolving shear bands, which are narrow regions of intense plasticdeformation that form in high strainrate deformation of thermally softening viscoplastic materials. Here we assume hat the
body s deformedn planestraincompression t a nominalstrainrateof 5000 ec
1,
and model a material defect by introducinga temperature perturbation at the center of the block.
1 IntroductionIn nearly all of the previous numerical studies of shear bands in twodimensional problemsinvolving a viscoplastic material (e.g., see Needleman 1989; Batra and Liu 1989; Batra andZhu 1991),a fixed finite element mesh has been used. Since shear bands are narrow regions ofintenseplastic deformation, their satisfactory esolution requireseither a very fine mesh hroughout the computational domain, in which case he solution in most of the domain outside he shearband is overcomputed, or an adaptively refined mesh that concentrates more elements n theseverelydeforming region and fewer elementsoutside of it. Batra and Kim (1990)developedan
adaptive mesh refinement technique or the analysis of onedimensionalshear banding problemsby ensuring that the scaled esidualsof the equations expressing he balanceof linear momentumand the balance of internal energy were uniformly distributed. They subdivided elementshavinglarge s~led residuals and observed hat high values of the scaled esiduals occurred, n general,in nonoverlapping regions. Their technique did not combine elementswith low values of thescaled esiduals,and for this reasondid not result in an optimum mesh.We make no attempt to
review all of the literature on adaptive mesh refinement and twodimensional adiabatic shearbanding problems. For the former, we refer he reader to Safjanet al. (1991)and Zienkiewicz andZhu (1991),and for the latter to Batra and Zhu (1991).2 Formulation of the problem
We use a fixed set of rectangular Cartesian coordinates with srcin at the centroid of a squareblock (cf. Fig. 1) to analyze its plane strain thermomechanical deformations. We assume that theblock is made of a thermally softening viscoplastic material. In terms of the refefential description,governing equations are:(pJ).
= 0,
POVj
= Tja.a, poe = Qa.a + TiaVi,a, (13)whereJ=detFia, Fja = Xj,a' Xi,a=OXj/oXa, (4)Xj is the present location of a material particle that occupied place X" in the reference configuration.
Computational Mechanics 10 (1992)70
tX2.X2
x"x,
Fig. 1. A schematic sketch of the problem studied
p its presentmassdensity, Po ts massdensity n the reference onfiguration, Vi ts presentvelocity,
Ti2 the first PiolaKirchoff stress ensor, Qa he heat flux per unit referencearea, e the specificenergy,a superimposeddot indicates he material time derivative, and a repeated ndex impliessummation over the range of the index. For the constitutive relations we take
0"..=
B
(
E1
)
<5..+2
1
D.. T. =' ~ "(5152)) I) I)' la . , .Po
Ol)x.0..«,) IJ

p
(6)
(J)
Jl = *(1 + b1)m(1...;31
2

2D..=v..+v.. 21 =D..D..
I) I,J },I' I} I}'
lD
kk
b..
3 'i
(7.17.3)
..=D..
IJ IJ
(8.1,8.2)
8.
,I
a
= ~Xa,jqj,
p
qi
.
(
p
)
p
=
cO B   1 . (9)
Po
(Pop)Here O"j.s the Cauchy stress ensor, B may be thought of as the bulk modulus for the material ofthe blo~k, D is the strainrate tensor, 0"0 he yield stressof the material in a quasistatic simpletension or compression est, parameters band m characterize he strai!!rate sensitivity of thematerial, I is the second nvariant of the deviatoric strainrate tensor D, v is the coefficient ofthermal softening,k equals he thermal conductivity of the material, c the specificheat, and 0 thetemperature rise of a material particle.We introduce nondimensionalvariables, ndicated below by a superimposedbar, as follows.f = tyo, J = l/yo, b = byo, Ii = p/Po, i/= 0'/0"0' t = T/O"o,B=B/O"o, v=vO" (f=O/O" v=v/vo, i=x/H, X=X/H, (lOa)
{) = Pov~/O"o, P = k/(PocvoH),
where0, = O"o/(Poc),Yo= vo/H. (lOb)In Eq. (10)2H is the height of the squareblock, 0, the referenceemperature,Vo s the steadyvalue
of the velocity applied to the top and bottom surfacesn the x2direction, and Yoequals he averageapplied strainrate.Henceforthwe usenondimensionalvariablesand drop the superimposed ars.
We presume hat the deformationsof the block are symmetricalabout the horizontal and verticalcentroidal axes,and study the deformations of the material in the first quadrant.For the boundary conditions we takeV2=0, TI2=0, Q2=0 on X2=X2=0, (11.1)VI =0, T'1 =0, QI =0 on Xl =XI =0, (11.2)
R. C. Batra and K.I. Ko: An adaptive mesh refinement technique for the analysis of shear bands
T" =0,
. .
on
.
, (11.3)
v2=h(t), T'2=0, Q2=0 on X2=H. (11.4)The boundary conditions (11) signify that the boundaries of the block are insulated, the rightsurface s traction free, here s no tangential traction acting on the top surface,and the top surfacemovesdownward at a prescribedspeedh(t). The boundary conditions (11.1)and (11.2) ollow from
the assumedsymmetry of deformations about the X, and X 2 axes.For the initial conditions we takep(x,O) = 1, Vl(X,O)= 0.37xl' V2(X,0)  x2, (12.112.3)()(x,O)= 0.2(1 r2)geSr2, r2= X: + X~. (12.4,12.5)The initial conditions on the velocity field represent he situation when the transients have diedout. Batra and Liu (1989) ound this velocity field by takingh(t)
= tjO.005, 0 ~ t ~ 0.005,
= 1, t ~ 0.005,assuming hat the initial temperature distribution is uniform, and computing the solution till thesteady state had been reached.The changes n the massdensity and the computed temperaturerise were found to be nsignificant to justify assuming hat the initial massdensity is uniform. Theassumptions (12.2) and (12.3) result in a smaller value of the CPU time needed o analyze theproblem and do not affect he qualitative nature of the results.The initial temperature distributiongiven by (12.4)modelsa material inhomogeneity; he amplitude of the perturbation can be thoughtof as representing he strength of the singularity.Equations obtained by combining (1) through (9) are to be solved under the side conditions(11) and (12). Since thesecoupled equations are highly nonlinear, it is not clear whether or notthey have a unique solution. Here we find their approximate solution by first reducing the partialdifferential equations to a set of coupled, nonlinear, and ordinary stiff differential equations byusing the Galerkin approximation. The number of theseequationsequals our times the numberof nodes in the finite element discretization of the domain. We use threenoded isoparametrictriangular elementsand the lumped massmatrix obtained by the rowsum technique. These stiffordinary differential equationsare ntegrated with respect o time by using he backward differenceAdam's method included n the subroutine LSODE (e.g., eeHindmarsh 1971).We could not usethe Gear method because f the limited core storageavailable o us.The computer codedeveloped
by Batra and Liu (1989)was suitably modified to solve the presentproblem.
f'Jl =0,
O. =0X.=H
3 Adaptive mesh efinement echniqueWe first selecta coarsemeshand find a solution of the aforestatedproblem. This mesh s refined
so that
ae= J Idn, e= 1,2,...,ne" (13)n.
is nearly the same or each element ne. In (13), nel equals he number of elements n the coarsemeshand ne is one of the elements.Sinceone may not havean idea where he solution will exhibit
sharp gradients, we choose he coarsemesh o be uniform. The motivation behind making ae hesameover eachelementne is that within the region of localization of the defonnation values of I
are very high ascompared o those n the remaining region.Other variablessuch as the temperature
rise, he maximum principal strain, and the equivalent strain which are also quite large within theband will be suitable replacementsor I in Eq. (13).The refinedmeshwill depend upon the variable
used n Eq. (13). n order to refine the mesh,we find
1 "., a
Ii
1 N.a
= '""'a
;: =!.
h
=!.
~
ndH
=
~
hn
=
1211 e' ..e ' e ,  " L.. e' , ,..., 04' ,,nele=l a c.e Nee=l
(1417)
Computational Mechanics 10 (1992)
72
Here, h" is the sizeof the element l" in the coarsemesh,N eequals he number of elementsmeeting
at node n, and nodequals the number of nodes n the coarse mesh. We refer to Hn as the nodalelementsize at node II.In order to generate he new mesh,we first discretize he boundary by following the proceduregiven by Cescotto and Zhou (1989).Let AB be a segmentof the contour to be discretized,s the
arc lengthmeasuredrom point A, and H
A
and H
B
be nodal element izes or nodes ocatedat
points A and B, respectively.From a knowledgeof the valuesof Hat discrt:tepoints, corresponding
to the nodes n the coarsemesh,on AB we define a piecewiseinear continuous function H(s) that
takes the previously computed values at the node points. In order to discretize AB for the newmesh,we start from point A if H
A
< H
B;
otherwise we start from B. For the sake of discussion,let us assume hat A is the starting point. We first find temporary positions of nodeson the segmentAB by using the following recursive procedure. Assume hat points 1,2,..., k have been found.Then the temporary location of point (k + 1) s given by
Sk+ 1
=
Sk
+
t[H(Sk)
+
H(s:+ 1)]' (18)
where
S:+ 1
=
Sk
+
H(Sk). (19)
Referring to Fig. 2, the above procedure will give rise to the following four alternatives: a
=
b
= 0,a < b, a > b, a = b # O. f a = b = 0, then the temporary locations of node points are their finalpositions. Depending upon whether a < b or b ~ a, the node points 2 to p or 2 to p + 1 are mov.ed,the displacementof a node being proportional to the value of H there, so that either node p ornode (p + 1) coincides with B. This determines he final positions of nodes on the segmentAB.Having discretized the boundary, we use the concept of advancing front (e.g.,see Lo 1985;Peraire et al. 1987, 1988;Habraken and Cescotto 1990) o generate he elements.An advancingfront consistsof straight line segmentswhich are available o form a side of an element.Thus, to
start with, it consists of the discretized boundary. We choose he smallest ine segment say side
AB) connecting the two adjoining nodes, and determine the nodal element size HM = H(SM)
=
(HA
+
HB)/2 at the midpoint M of AB. We setr 0.8AB ~
if
{)HM1.4AB

HM<O.8AB,
 if O.8AB ~ HM ~ 1.4AB,
if 1.4AB < HM,and find point C1 at a distance ) from A and B (cf. Fig. 3). Here AB equals the length of segmentAB. We search or all nodes on the active front that lie inside the circle with center at C1 and
Fig. 2. Discretization f a boundary segment or mesh efinement
Active
Fig. 3. Advancing ront and new elementgeneration
373
. C. Batra and K..I. Ko: An adaptivemesh efinement echnique or the analysisof shearbands
radius fJ,and order them according to their distance rom Ct with the first node in the list beingclosest o Ct. At the end of this list are added points Ct, C2, C3, C4, and Cs, which lie on CtMand divide it into five equal parts. We next determine he first point C in the list that satisfies hefollowing three conditions.(i) Area of triangle ABC> O.(ii) SidesAC and BC do not cut any of the existing sides n the front.(iii) If any of the points Ct, C2,... , C 5 s chosen, hat point is not too close o the front.The triangle ABC is an element n the new mesh. f C is one of the points Ct, C2,..., Cs, then anew node is also created.The advancing ront is updated by removing the line segmentAB fromit, and adding line segmentsAC and CB to it. The elementgeneration processceaseswhen there
is no side eft in the active front.We determine the values of solution variables at a newly created node by first finding out towhich element n the coarsemesh his node belongs,and then finding values of solution variablesat this node by interpolation. This processand that of searching or line segmentsand points inthe aforestatedelementgeneration echniqueconsumea considerableamount ofCPU time. These
operations are optimized to someextent by using the heap ist algorithm (e.g.,seeLohner, 1988)
for deleting and inserting new ine segments, nd quadtreestructures and linked lists for searchingline segmentsand points and also for the interpolation of solution variables at the newly creatednodes.4 Resultsand discussionWe assume hat the block is made of a typical steeland assign he following values,also used byBatra and Liu (1989), o various parameters.b= 10,OOOsec,O"o=333MPa, k=49.2Wm1°C1, m=0.025, c=473Jkg1°C1,Po = 7,800kg/m3, B = 128GPa, (21)v=0.0222°C1, vo=25msec1, H=5mm, h(t) = 1.0.Here we have made an exception to our notation and indicated dimensional quantities to clarifythe units used.As statedearlier, the transients are assumed o have died out, the top surfacemoves
downward with the prescribed speedva' and the averagestrainrate at which the block is beingdeformed equals 5000secl. For values given in (21), Or= 89.6°C, and the nondimensional
melting temperature equals 0.5027.We note that the value of the thermal softening coefficient vhas been purposely taken to be high so as to reduce he computational time. It should not affectthe qualitative nature of the results reported herein. The test data to find values of materialparametersat strainrates,strains,and temperatures ikely to occur n a shearband s not available.
Figure 4 depicts the initial coarse mesh at time t = 0, and the generated refined meshesatnondimensional time t
= 0.025,0.040, nd 0.047.We note that the nondimensionalime also
equals he averagestrain. In the solution of the problem, the mesh was also adaptively refined att = 0.015,0.030,and 0.035;however, heseare not shown here for the sake of brevity. The times
at which the mesh s refined were selectedmanually, and are to some degreearbitrary. A possiblecriterion could be to refine the mesh when the second nvariant of the strainrate tensor or thetemperatureat the centerhas risen by a certain amount. The meshes hown n Fig. 4 vividly revealthat the refinement echnique outlined in Sect.3 gives rise to nonuniform mesheswith finer meshin the severelydeforming region and coarse mesh elsewhere.We did not impose any restrictionon the number of new nodes hat can be ntroduced when the mesh s refined. Practical considerations such as the core storage available may require this kind of restriction.In Fig. 5 we have plotted the contours of the second nvariant I of the deviatoric strainratetensor at t = 0.019,0.032,0.042,and 0.047 n the deformed configuration. Theseplots suggest hat
as the block continues to be deformed, he deformation localizes nto a band whose width keepson decreasing.Contours of successivelyncreasingvaluesof I srcinate from the centerof the block
and propagate outward. The contours of the temperature rise at t
= 0.019, .032, .042, nd 0.047