finite element simulation of wedge indentation

Compurw & Swurrures Vol. 57, No. 5. pp. 915-927, 1995

0045-7949(94)00557-5 Copyright e:: 1995 Elsevicr Science Ltd

Printed in Great Britain. All rights reserved 0045-7949195 s9.50 + 0.00

FINITE ELEMENT SIMULATION OF WEDGE

INDENTATION

K. R. Jayadevant and R. Narasimhan$

TDepartment of Mechanical Engineering, Government Engineering College, Thrissur, India

IDepartment of Mechanical Engineering, Indian Institute of Science, Bangalore, India

(Received 21 February 1994)

Abstract-A finite element simulation of frictionless wedge indentation of a copper strip has been carried out under plane strain conditions. The problem was first modelled using an one-pass contact algorithm. The difficulties associated with using this method to model wedge indentation problems are explained. An alternative procedure which alleviates some of the problems associated with the one-pass contact algorithm is proposed for modelling frictionless wedge indentation. Also, a re-meshing procedure which has to be carried out when the distortion of the elements around the indenter becomes significant, is discussed. A sample problem involving indentation of a 4 mm copper strip by a rigid wedge indenter has been modelled and the results are compared with experimental and theoretical results.

1. INTRODUCTION

Indentation hardness testing is widely used as a technique to determine the strength of materials. Hill et al. [l] constructed an analytical solution for a frictionless wedge indenting a rigid-perfect plastic half-space using two-dimensional plane strain, slip- line theory. This solution assumes that the shape of the indented zone is geometrically similar irrespective of the depth of indentation. Thus, the flow pattern around the indenter and the pressure exerted by the indenter on the surrounding material remain unaltered with depth of indentation.

Dugdale [2] conducted wedge indentation experiments on cold-worked metals and found that the measured hardness is independent of size, provided that the metal blocks are larger than a critical size. He analysed the hardness data using the slip-line field of Hill et al. [l] and concluded that the yield stress (in shear) extracted from the indentation test compares well with that obtained independently from torsion tests. The work of Dugdale [2] established firmly the use of wedge indentation as an experimental technique for measuring the strength of metals.

In more recent years, depth sensing indentation has been applied [3-61 to obtain the properties of layered materials and thin films which are employed in high technological applications. Thus, properties of thin films such as hardness [3], elastic modulus [4], stress-strain curves [5] and adhesion [6] have been extracted from results of indentation experiments. However, these studies are based mainly on empirical analysis of test results rather than on a comprehen- sive analysis based on the theory of plasticity. The finite element method which is ideally suited for conducting such studies has been employed by a

few investigators [7, 81 for analysing large indenta- tions by a ball indenter which is typical of the Brine11 hardness test. Komvopoulos [9] conducted a finite element analysis of the indentation (by a rigid cylindrical surface) of an elastic-plastic layered solid. Bhattacharya and Nix [lo] have performed elastic- plastic finite element simulations of axi-symmetric indentation by a conical indenter.

In conducting a finite element simulation of indentation, contact algorithms can be employed. The most popular among these are the node-to-segment slide-line methods [ 1 l-141. Hallquist [ 1 l] proposed a simple procedure for numerical treatment of contact phenomena under impact conditions using sliding interfaces. Bathe and Chaudhary [12] developed a solution procedure for modelling axi-symmetric and two-dimensional (plane-strain and plane-stress) contact between two bodies using the Lagrange multiplier method. Wriggers et al. [13] have presented a procedure for analysing contact-impact problems taking into account large deformations and a class of general friction laws. Taylor and Papadopoulos [ 141 have proposed a patch test for contact problems. They conclude that a two-pass algorithm which enforces the contact constraints on both the bodies is more robust than an one-pass algorithm. However, it should be noted that a two-pass algorithm is much more difficult to implement than an one-pass algorithm.

The objective of this work is to develop a finite element procedure for simulating indentation of a metal strip by a rigid wedge type indenter under the conditions of two-dimensional plane strain. A finite deformation elastic-plastic constitutive model obeying the J, flow theory of plasticity along with an updated Lagrangian finite element formulation is

915

916 K. R. Jayadevan and R. Narasimhan

used in this work. For modelling wedge indentation, a general node-to-surface contact algorithm [ 12, 131 is first tried. Since this gives rise to some numerical difficulties, a new method is developed to simulate frictionless wedge indentation. Also, mesh re-zoning is carried out in order to continue the analysis for large deformations because of the considerable distortion of the finite element mesh. The above finite element procedures have been implemented in the general purpose finite element code FEAP [15]. A sample problem pertaining to indentation of a copper strip was modelled using the proposed procedure to test its performance. The re-zoning of the mesh was carried out to continue the analysis to a large indentation depth. The numerical results are compared with theoretical and experimental results.

2. FINITE ELEMENT PROCEDURE FOR FINITE STRAIN PLASTICITY

Elastic-plastic indentation of metals by a wedge indenter introduces large strains and also causes large rotation of material elements near the tip of the indenter. Thus, the problem involves material and geometric non-linearity. The constitutive equations of plasticity lead to material non-linearity and the treatment of finite strains and rotations introduces geometric non-linearity.

In the present work, an up&ted Lagrangiun formulation (see Ref. [16]) is used in order to treat finite deformation. Attention will be restricted to static analysis and to materials obeying the J1 flow theory of plasticity with small elastic strains. A mean dilata- tion formulation [17] is employed, since the deformation becomes nearly incompressible in the fully plastic range which cannot be adequately treated by the conventional displacement method. Finally, the Hughes and Winget algorithm [18] is used in this work to treat finite rotation effects, so that incremental objectivity is maintained during constitutive update.

2.1. Finite element equations

In the updated Lagrangian procedure, a reference configuration which coincides instantaneously with the current configuration is chosen, so that the Kirchoff stress TV, and the Cauchy stress 0,) are the same, whereas their rates are different. In particular, the Jaumann rate of the Kirchoff and Cauchy stress tensors are related by

r; = 0:: + aiiDk,. (1)

By considering the static equilibrium of a body at time t and t + At, and invoking the virtual work rate principle, the following variational equation can be derived [ 161

+ At s r,,v,,Sv,,, dV = At s

d,Sv, dV

+Ati;$&+d,4 +,jl,,b,&;dI’

+{Srqao,da -[V,n,bD,dV]. (2)

In this equation, all integrations are performed over the volume I’, and the boundary segment S, in the equilibrium configuration corresponding to time t, and ak,, = &,/ax, is the spatial gradient of particle velocity vector. Also, 6u, is a virtual velocity field which is imposed on the current equilibrium configuration and 6D,, = f(&,,, + Su,.,) denotes the associated virtual rate of deformation tensor. Further, 6, and p, represent nominal body force and surface traction rates based on the current volume and surface area. It should be noted that eqn (2) applies strictly in a rate sense, as envisaged by McMeeking and Rice [ 161 and, thus, the time step size At is viewed here to be vanishingly small. The term in the square bracket on the right hand side of eqn (2) is an equilibrium correction term that vanishes when the known state at time t is an exact equilibrium state.

In this work, rate independent elastic-plastic constitutive equations which satisfy material objectivity are considered and are assumed to have the following structure:

r ; = M,,,,& 3 (3)

where, M,,kl is the elastic-plastic constitutive tensor. The constitutive tensor M,,k, for materials obeying a finite strain version of the JZ flow theory of plasticity (with small elastic strains) along with isotropic strain hardening is given in Ref. [ 161.

In the updated Lagrangian procedure, a finite element discretization of the body in the current configuration is considered and eqn (2) is written as a summation over the elements. The finite element equations for rate equilibrium can be derived from eqn (2) by following the usual finite element procedure as (see Ref. [16]):

At’[K]{ti} = At{fi*} + At{F,}

Here, ‘[K] represents the tangent stiffness matrix corresponding to time t (see Ref. [16]), { ri} is the vector of nodal point velocities, {fib} and {$,‘I> are nodal force rate vectors due to body forces and surface tractions. Further ‘{ Fb } and ‘{F, } are external force vectors due to body forces and surface tractions at time t, and ‘{P} is the corresponding internal force

Simulation of wedge indentation 917

vector due to element stresses. The last three terms in the right hand side of eqn (4) are added to correct for deviations from the equilibrium state during incremental loading of the body.

In rate independent plasticity (where the material does not possess a natural time scale), the rate equilibrium eqn (4) is integrated in an incremental manner following a loading path.

3. FINITE ELEMENT MODELLING OF FRICTIONLESS SLIDING CONTACT

A general procedure for analysis of frictionless contact problems [12, 131 is discussed below in Section 3.1. This procedure is then applied to solve a problem involving wedge indentation of a copper strip in Section 3.2.

3.1. Finite element formulation

The node-to-surface slide-line method for modelling contact problems is briefly presented here. One of the two bodies which are in contact, is arbitrarily named as contactor and the other as target. In the finite element solution, the boundary nodes of the contactor that may come into contact with target boundary segments or nodes are declared as slave nodes and those of the target are called master nodes. The set of slave nodes collectively defines the slave slide-line and that of the master nodes, the master slide-line.

The basic condition, to be satisfied along the contact interface, is that there should not be any material overlap (impenetrability condition). Also, since the contact is frictionless, the tangential contact

forces along the contact surface should be zero. In the finite element formulation, the normal contact forces are generated due to the imposed contact constraints. A one-pass algorithm (using a penalty parameter), which imposes the impenetrability condition only on the slave nodes, is employed in this work. In the following development, and in the remaining sections, attention will be restricted to plane strain conditions and only four-noded quadrilateral elements will be considered.

For the formulation of the contact solution algorithm, an incremental procedure is followed. The virtual work rate due to the contact forces arising from the imposed boundary constraints for all the slave nodes in contact, is added to eqn (2). A generic slave node k which has come into contact with the target segment formed by nodes A and B is illustrated in Fig. 1. The vectors xk, xA and xB represent the position vectors of the nodes k, A, and B respectively, in the current configuration from the origin 0. Also, d is the length of the master segment AB. Further, Sk is the normal penetration of the slave node k into the target body. This penetration is first corrected by the contact algorithm (i.e. node k is first brought back to point C in Fig. 1). In the subsequent load steps, node k is constrained to lie on segment AB till a release due to a normal force directed away from the contactor body occurs. Another possibility that could arise is the sliding of node k beyond the target segment AB to a neighbouring target segment. Finally, p in Fig. 1 is a parameter which defines the position of the slave node k with respect to the target segment AB. The incremental displacements at nodes k, A and B during the time step from t to t + At are denoted by Auk, AuA and AuB, respectively (see Fig. 1). The (normal)

‘\ ‘, k

‘,,A” n

‘\ ‘.

‘\

Contactor

Fig. 1. Contact of two bodies showing geometric variables.


impenetrability condition for the slave node k during the incremental step from t to t + Ar is given by,

(Auk,+6:)-(I -fi)Au;f--pAu:=O, (5)

where Au%, Au!, and AU! are the components of the displacement increments of nodes k, A, and B normal to segment AB (see Fig. lb). In writing the above equation, the change in fi from time t to r + At, which will give rise to a term of higher order (see Ref. [13]) is not taken into account.

In the penalty formulation, the incremental contact force AL: exerted by the slave node k on the master segment AB in the direction normal to it. is identified to be the product of a penalty parameter r (a large positive number) and the above incremental constraint equation. Thus,

Aif,=r((Auj:+6;)-(1 -~)Au;-[{Au,“). (6)

The virtual work rate due to the above contact forces when a virtual velocity field 60 is imposed at time t + At is given by,

SW,,,, = F (‘j.: + a((Aui + S”,) - (I - p) k=l

where, N, is the number of slave nodes and ‘1: is the existing normal force (i.e. corresponding to time t) at node k. In writing the above equation, variation in p and in the normal direction n to the master segment AB [13] are not taken into account. It was found that the inclusion of these additional terms to the virtual work rate, eqn (71, and hence to the stiffness matrix did not affect the results of the modelling for wedge indentation problems. The interested reader may refer to Ref. [13] for the contribution to the stiffness matrix from the above virtual work rate.

3.2. Modelling of wedge indentation

An attempt was made to model the indentation of a copper strip of thickness 4mm resting on a frictionless surface by a rigid wedge indenter with an included angle of 90” using the above contact algorithm. Experiments were also conducted on the same geometry by Jayadevan [ 191. The inherent difficulties which were encountered in this numerical modelling are outlined here.

The rigid wedge indenter was modelled as the contactor body with a fictitiously large Young’s modulus and the copper strip was modelled as the target body. Thus the material properties used for the indenter material are: E (Young’s modulus) = IO' GPa and v (Poisson’s ratio) = 0.3. For the copper strip, E was assumed as 100 GPa and v as 0.3. Further, its initial yield stress ran was taken as 200 MPa and very low linear isotropic hardening was assumed with a tangent modulus E, = 10 MPa. Thus, the copper strip was assumed to be almost perfectly plastic. The penalty number a was taken as lOI* MPa. A coarse representation of the mesh layout used is displayed schematically in Fig. 2 with the specified boundary conditions. The dimensions of the strip and the indenter are also marked in Fig. 2. Taking symmetry into consideration, half of the problem was modelled with plane strain four-noded quadrilateral elements. In order to limit the computation time, the mesh was graded horizontally, with fine elements near the tip of the indenter. A detailed view of the deformed mesh near the tip of the indenter at two different indentation depths are displayed in Fig. 3a and b.

The most serious problem encountered in this analysis was the kinking of the master slide-line (see line ABCD in Fig. 3b). It was not possible to maintain the master slide-line as perfectly straight and in contact with the indenting wedge. This led to the decrease in contact forces at some slave nodes and

I ,I.smm_,

w f 1.5mm

Fig. 2. A coarse representation of the mesh layout used for modelling wedge indentation

919

DSF = O.lOOE+Ol Time = O.lOOE+O4

Simulation of wedge indentation

DSF = O.lOOE+Ol Time = O.l5OE+O4

(a) (b)

Fig. 3. Detailed views of deformed mesh near the indenter tip at an indentation depth of (a) 0.25 mm and (b) 0.3 mm.

to their eventual separation from the master slide-line (see, for example, the slave node k in Fig. 3b). It can also be observed in Fig. 3b that some master nodes (see for example, node C in Fig. 3b) penetrate into the contactor body. In fact, the penetration of master nodes into the contactor body is an inherent limita- tion of the one-pass contact algorithm (see discussion in Ref. [14]).

Another difficulty in the modelling was associated with the conversion of the boundary value problem from that pertaining to point contact at the interface node on the symmetry axis (see node A in Fig. 3b), to that pertaining to wedge indentation, when the second slave node on the slave slide line comes into contact with the master slide line. Because of the above mentioned change in boundary value problem, a jump in the initial portion of the load-displacement curve was observed, which can be smoothened to some extent by using very fine elements near the tip of the indenter. A third issue which was encountered was the jump in the load, every time a new slave node contacted the master slide-line. This created discontinuities in the load vs indentation depth curve.

In order to alleviate some of the above difficulties, an alternative procedure was developed in this work to model frictionless wedge indentation. This procedure, which obviates the need to explicitly model the rigid wedge, is discussed in the next section.

4. AN ALTERNATE PROCEDURE FOR MODELLING WEDGE INDENTATION

An alternate procedure which was developed to model the loading part of frictionless wedge indentation is presented in this section. The basic idea behind this formulation is the fact that when a material is indented by a wedge type indenter, the material particles on the contact surface should lie on the surface defined by the geometry of the indenter. In other words, there should be normal displacement compatibility between the wedge and the indented material. The relative tangential displacement between material particles on the either side of the contact surface will be dictated by the frictional condition at the interface. Since a frictionless condition is assumed in the pressent work, no constraint is imposed on the tangential displacements at the interface, and the tangential component of the contact force is taken as zero. Also, since the wedge is assumed to be rigid, it is not necessary to model the wedge in this simulation. Instead, the vertical downward displacement which is specified at the top surface of the wedge can be used directly to prescribe the normal component of the displacement at the nodes lying on the surface of the indented material which are presumably in contact with the wedge. As in Section 3.1, the above constraint equation is imposed via the penalty formulation.


Figure 4 shows the boundary nodes on the interface of the deformed material at any intermediate time t during the indentation analysis. Taking symmetry into account, only half the problem is considered, as shown in Fig. 4. Here, e, , e2 denote fixed Cartesian base vectors. Further, the boundary nodes on the interface are numbered here locally starting from the interface node on the symmetry line and proceeding towards the left (see Fig. 4). The specified vertical downward displacement increment given to the first node which will be in contact with the vertex of the wedge is denoted by Au:. Also, 0 represents the semi-wedge angle (see Fig. 4). The constraint equation for the nodes lying on the boundary segments which have already turned to angle 8 with the vertical axis and, therefore, are in contact with the wedge is given by:

Aui=C, fork=2,3 ,..., I-1. (8) 6 IV,,,, = s((AuI, - d’- ’ sin tj - C)Gvf,

Here, AU; is the displacement increment of node k (local node number) in the direction of the normal n in Fig. 4 and C = AU: sin 8. Further, the local node number of the last node which is in contact with the wedge at time t - At, where, At is the time step size in the incremental analysis, is I - 1.

In the following development, it is assumed that only one new segment [i.e. segment (I - l)] will rotate and may go beyond the inclined wedge surface during the incremental time step from t -At to t. In other words, the possibility of only node I in the Fig. 4 penetrating the wedge between time step t -At to t is considered. In that case, such a node I will have to be constrained accordingly to remain on the inclined surface during the next time step from t to t + At and further on in the analysis. The restriction regarding the rotation of only one additional node (i.e. node I in the present context) to the wedge surface during an incremental step is consistent with the requirement

--__ I+1 1

q (I-l)

(I-l) d 3 ’ ! w el

G! I-1 (I-2)

I-2 ‘. \ n /e2 ‘.

3

< (2) 8 ’

2 ! Cl)\

I’

I

Fig. 4. A schematic of the deformed surface of the indented material showing the boundary nodes which would make

contact with the indenter.

that the displacement increment specified during any incremental time step should be very small and should at most be a small fraction of the element size. Hence, the constraint equation for the node I for the time step t to t + At becomes:

Au: = (d”- ‘) sin $) + C. (9)

In the above equation, d(‘- ‘1 is the length of the segment (I - 1) as shown in Fig. 4 and II, is the angle turned by that segment over and above the semi- wedge angle 8. On imposing the above constraint eqns (8) and (9) using the penalty method, the virtual work rate due to the contact forces, at the wedge surface when a virtual velocity field 6v is applied on the equilibrium configuration at time t + At can be written as,

I-I

+ c (‘1; + cc(Au: - C))&$ (10) k=2

In the above equation, a is a large positive number (penalty number) and ‘d% is the existing (i.e. corresponding to time t) contact force in the direction of the normal n (see Fig. 4) at node k exerted by the indented material on the wedge surface. Hence, ‘1: in the present context will have a negative value. The normal and tangential displacement increments are related to the corresponding Cartesian components with respect to e, and e, through the rotation matrix Q as:

{;;;} = [Ql{;::}. (11)

In the above equation (see Fig. 4)

[Q] = L zIs; -case . sin e I (12)

Thus, due to the kth interface node (k E [2, I - l]), the contribution to the stiffness matrix in the left hand side of the incremental equilibrium eqn (4) is given by,

[K&J = a [ cos2 e cos e sin e sin e cos e sin2 e 1 (13)

The contribution to the force vector due to the above node k in the right hand side of eqn (4) is as follows:

Vtm,J = (14)

The contribution due to the newly rotated node I (see Fig. 4) to the stiffness matrix is the same as that given by eqn (13) above. On the other hand, its contribution to the force vector will be as follows:


{Ffon, 1 = u (d(‘- ‘) sin I++ + C)cos ~9

(d(‘-‘)sin$ +C)sin6 . (1%

After solving the incremental equilibrium eqn (4), the normal contact forces are updated as follows:

‘+A’l~=rL~+tl(A~~-C) forko[l,I-11, (16)

and

‘+“‘A: = a(Auf, - C - d(‘-‘) sin $). (17)

It should be noted again that physically the above quantities represent the forces in the direction of normal n (see Fig. 4) exerted by the indented material on the wedge (which, however, is not simulated explicitly in this formulation).

After updating the configuration, and before proceeding to the next incremental step (from t + At to t + 2At), a node state evaluation is done, to see whether the next node I + 1 (see Fig. 4) has pene- trated the wedge interface. The performance of this procedure will be discussed in Section 6.

5. MESH RE-ZONING

The performance of isoparametric elements is generally best when they are regular [20] in shape and retain the geometry of the parent elements. However, the elements which are initially rectangular or triangular may become highly distorted in a geometrically non-linear analysis, particularly when it pertains to applications such as metal forming, indentation, impact, etc. Finally, a stage may be reached when one of the edges of an element in the distorted zone may collapse to a point, or one of the internal angles of an element may exceed 180”. This makes further analysis impossible, because the Jacobian of the isoparametric mapping for such an element will become negative. Also, it has to be noted that geometric distortion stiffens the element and makes the finite element solution to the boundary value problem less accurate [20]. In the finite element modelling of wedge indentation, the elements in the indented material close to the vertex of the wedge will undergo severe distortion. Thus, it becomes necessary to monitor the shape of the elements in this region during the analysis and a re-zoning of the mesh should be performed when the distortions become significant.

The re-meshing operation involves the following successive steps:

(1) the re-meshing decision, (2) the new mesh generation, and (3) the transfer of information from the old mesh

to the new mesh.

In the present work, the re-meshing decision was arrived at purely on a qualitative basis by visually examining the deformation of the elements in the

finite element grid near the vertex of the wedge. A more systematic way of deciding on the stage for re-zoning can be based on error estimation techniques [21]. The basic idea is that re-meshing should be done when the error due to spatial discretization (connected with the finite element interpolation) becomes significant.

After taking the re-meshing decision, the steps adopted in the mesh re-zoning technique are as follows:

(1) generate nodal values of all stress components and plastic internal variables such as current yield stress and equivalent plastic strain in the old (distorted) mesh, from the corresponding values at the element Gauss quadrature points, using a least- squares smoothing technique [22];

(2) generate a smooth mesh in the distorted region, adjusting the spacing of the boundary nodes and moving the inner nodes, and using the displaced positions of the boundary nodes in the old mesh to define the domain of the new mesh;

(3) locate all Gauss quadrature points of each element (both 2 x 2 and 1 x 1 Gauss points) of the new mesh within the framework of the old mesh;

(4) after locating a Gauss point belonging to the new mesh inside an element of the old mesh, determine its natural co-ordinates ([, ‘I) with respect to that element of the old mesh;

(5) obtain the values of all stress components and plastic internal variables like current yield stress and equivalent plastic strain at all Gauss points of each element of the new mesh using the interpolation formulae along with the smoothed nodal values of the old mesh. Here, plastic consistency is enforced by a simple radial return method,

An efficient procedure for performing steps (3) and (4) is described in Appendix A.

In wedge indentation analysis, the following additional steps should be adopted to determine the contact forces at the new nodes on the inclined surface after mesh re-zoning:

(1) compute the traction distribution on the wedge interface in the old mesh from the contact force values at the nodes on the inclined surface, using an inverse of the consistent nodal method [20];

(2) obtain the magnitude of the tractions on the wedge interface at the nodes belonging to the new mesh by employing the traction distribution determined in the previous step. A simple interpolation can be used for this purpose;

(3) compute the nodal force values at the interface nodes in the new mesh from the corresponding traction distribution, using the consistent nodal approach.

From some numerical experimentation, it was found that just after re-zoning, the stress variation within the elements of the new mesh may not satisfy equilibrium because these stresses are generated by


least-squares smoothing of the discrete element stresses of the old mesh. The finite element system can be brought to equilibrium by using a few very small load steps just following the re-zoning.

6. RESULTS AND DISCUSSION

The indentation of a copper strip of 4mm thickness by a frictionless rigid indenter having a semi- wedge angle of 45”, which was described in Section 3 (see Fig. 2) was simulated again using the procedure presented in Section 4. The dimensions of the copper strip and the boundary conditions (with only half the strip through the wedge axis being considered because of symmetry) are the same as in Fig. 2. Also, the material properties for the copper strip, as well as the value of penalty number o! are taken to be the same as in Section 3. The essential difference is that, unlike in Section 3, the (rigid) wedge is not modelled here, and only the copper strip is discretized by finite elements. Further, the re-zoning method described in Section 5 was employed to re-zone the mesh locally around the indenter, whenever the distortion of the elements in this region became large. This facilitated the use of very fine elements near the indenter tip.

In Fig. 5a, a portion of the undeformed mesh used at the beginning of the simulation to model the copper strip is displayed. All elements used in this analysis are four-noded isoparametric (plane strain) quadrilateral elements. The vertical boundary on the right side in this figure is the symmetry line through the wedge axis and the top horizontal boundary is the free surface of the copper strip. Thus the tip of the wedge will come into contact with the node at the intersection of these two boundaries. In Fig. 5b, the deformed mesh near the indenter tip when the indentation depth is 0.03 mm is shown. Here, the inclined

top edge of the corner element has not yet rotated to an angle of 45& with respect to the vertical axis. The incipient formation of the lip can be observed from the deformation of the top edge of the element adjacent to the corner element.

At the stage represented in Fig. 5b, the mesh was locally re-zoned around the indenter. The re-zoned mesh in which additional elements have been intro- duced near the vertical symmetry axis is shown in Fig. 6a. One reason for introducing these additional elements is to suppress the jump in the load which would occur when an element edge on the inclined top surface of the strip comes into contact with the wedge surface. It should be mentioned here that additional constraint equations were used to enforce displacement compatibility among the nodes lying along the line AA in Fig. 6a which separates the re-zoned portion from the remaining part of the mesh. The analysis was then continued using the mesh as shown in Fig. 6a till an indentation depth of 0.105 mm was attained. The deformed mesh at this stage is presented in Fig. 6b.

DSF = O.lOOE+Ol Time = O.OI_WE+OO

DSF = O.lOOE+Ol Time = 0.3OOE+O3

(b)

Fig. S.(a) Undeformed mesh showing the region modelled near the indenter. (b) Deformed mesh at an indentation

depth of 0.03 mm.

It can be seen on comparing Fig. 6a-b that large deformations involving rotation and distortion of elements have taken place mainly around the indenter tip. The remaining elements (away from the indenter) have undergone very little deformation. Further, it can be observed from Fig. 6b that at this stage (indentation depth = 0.105 mm) the lip has become prominent and encompasses the top edge of several elements adjacent to the free surface. The angle 4 made by the lip with a horizontal line as indicated in Fig. 6b is around 12”. This is in close agreement with the theoretical results of Hill et al. [I] for a wedge indenter having an included angle of 90”. It should be observed from Fig. 6b that the inclined surface of the indented strip (which is in contact with the wedge) remains perfectly straight. There is absolutely no kinking in the contact surface which was a major source of difficulty in the previous modelling using the contact algorithm (see Figs 3a and b).

The mesh was re-zoned locally around the indenter at the stage represented in Fig. 6b and the re-zoned


DSF = O.lOOE+Ol Time = O.OOOE+oO

DSF = O.OOOE+OO Time = 0.75OE+O3

(b)

DSF = 0.1OOE+01 Time = 0.75OE+O3

Fig. 6. (a) Re-zoned mesh at an indentation depth of 0.03 mm. (b) Deformed mesh at an indentation depth of

0.105 mm.

mesh is shown in Fig. 7a. The analysis was then continued up to an indentation depth of 0.18 mm. The deformed mesh corresponding to this stage is displayed in Fig. 7b. As in Fig. 6b, the inclined contact surface is observed to be straight with no kinking. Also a prominent lip with an angle 4 of about 12” with a horizontal line can be seen as indicated in Fig. 7b. The numerical analysis was stopped at this stage.

The variation of load (per unit thickness) vs indentation depth obtained from the numerical solution is plotted in Fig. 8. The experimental result of Jayadevan [19] is also shown in Fig. 8. In Ref. [19], experiments were conducted using the same geometry as in Fig. 2 with the out-of-plane thickness of the strip equal to 8mm. Thus, to facilitate comparison, the load obtained from the experiments was first divided by the above thickness and then plotted in Fig. 8. It can be seen from Fig. 8 that the numerical result compares well with the load vs indentation depth obtained from the experiments.

(a)

DSF = O.lOOE+Ol ‘lhe=O.75OE+O3 .A----

(b)

I I I I I I I I I I Fig. 7. (a) Re-zoned mesh at an indentation depth of 0.105 mm. (b) Deformed mesh at an indentation depth of

0.18 mm.

The numerical curve in Fig. 8 shows no discontinu- ity or kinks except in the initial part of the loading (around an indentation depth of 0.035 mm). This

______N”mefid

- Experimental

0 0.1 0.2 0.3

Fig.

Indentation depth (mm)

8. Comparison of load vs indentation depth obtained from numerical and experimental results [19].


kink is attributed to the conversion of the boundary Also, unlike in the previous simulation using the value problem from that pertaining to point contact contact algorithm (see Section 3) no decrease in the at the tip of the indenter, to that pertaining to wedge normal load at the nodes on the contact surface was indentation, when the first element edge (see Fig. Sb) observed. This is an outcome of the fact that the rotates to the angle of 45“ with the vertical axis. contact surface (see Figs 6b and 7b) remained straight

Time = 0.3OOOOE+O3 DSF = 0.1OCQOE+Ol

Min (-) = 7.85E-02 Max (+) = 2.05B+02

Contour levels (MPa)

A O.OOE+oO B %OOE+Ol C l.O0E+O2 D 1.5OE+O2 E 2.OOE+O2

(4

Time = 0.75OOOE+O3 Min (-) = 5.3OE-01 DSF = 0.10000E+01 Max (+) = 2.31E+02

Contour levels (MPa)

A 9.OOE+Ol B l.l5E+O2 C 1.4OE+O2 D 1.65E+O2 E 19OE+O2

Fig. 9. (a) Contours of equivalent stress at an indentation depth of 0.03 mm. (b) Contours of equivalent stress at an indentation depth of 0.18 mm.


during the present simulation and did not exhibit any kinks. The use of refined elements near the indenter, which was made possible by the re-zoning technique described in Section 5, has helped in suppressing the jump in the load when an element edge on the top surface of the strip comes into contact with the wedge surface.

It can be seen from Fig. 8 that the load increases monotonically and almost linearly with indentation depth. This is because of the fact that while the normal contact pressure exerted by the wedge on the strip is uniform (see Ref. [l]) and remains invariant with indentation depth (at least for small indentation depths), the length of the inclined contact surface increases linearly with indentation depth. The load vs indentation curves presented in Fig. 8 are about 10% higher than the theoretical variation based on the slip-line field of Hill et al. [l]. This difference is attributed to the fact that the slip-line field pertains to rigid-perfectly plastic materials. The strong strain concentration near the indenter tip has caused local elevation of the flow stress in this region (due to strain hardening) in the numerical simulation and in the experiments.

The contours of Mises equivalent stress o, are shown in Fig. 9a and b corresponding to the stage when the indentation depth is 0.105 and 0.18 mm, respectively. The contour levels (in MPa) are also indicated in the figures. In Fig. 9a, the contour marked as E (corresponding to 6, = 200 MPa) represents the boundary of the plastic zone. In Fig. 9b, contour E (which corresponds to (TV = 190 MPa) should be slightly larger than the elastic-plastic boundary. It is interesting to note that the contours of Q, spread sidewise (along the free surface) as well as below the indenter. In fact, this can be understood by examining the slip-line field of Hill et al. [l]. On comparing Fig. 9a and b, the growth of the plastic zone due to additional indentation in Fig. 9b can be observed.

7. CONCLUSIONS

The following are the important conclusions of this work.

(1) The modelling of plane strain wedge indentation by an one-pass contact algorithm leads to several numerical difficulties. The most important of these is the kinking of the contact surface.

(2) An alternate procedure has been developed for simulating frictionless indentation of a metal strip by a rigid wedge. In this procedure, the wedge is not modelled explicitly. Instead, the constraint equations are imposed directly on the nodes lying on the contact surface of the indented material.

(3) A method for re-zoning the mesh around the indenter (which is suited for plane strain four-noded quadrilateral elements) has also been discussed. Here, an efficient method for locating the Gauss

points of the new mesh within the elements of the old (distorted) mesh has been proposed.

(4) A sample problem involving frictionless indentation of a copper strip by a rigid indenter has been simulated with the above procedure. It is found that the numerical difficulties associated with the conventional contact algorithm are not exhibited in this procedure. Also, the numerical results compare well with analytical and experimental results.

(5) Finally, with the aid of the mesh re-zoning technique, it is possible to use refined elements near the indenter. Also, it is possible to continue the simulation for large depths of indentation. Thus, this technique can be applied to simulate indentation of adhesively bonded layered solids in which the mechanics of deformation has been found to change as the indenter penetrates deeper into the layer [19, 231.

Acknowledgements-The authors would like to thank the Supercomputer Education and Research Centre, IISc, Bangalore for providing the computational facilities to carry out this work. The assistance rendered by Mr T. S. Panishayee, Research Assistant, Mechanical Engineering Department, IISc is also acknowledged. The second author would like to gratefully acknowledge the Indian Space Research Organization for financial support through sponsored project ISTC/ME/RN/53.

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REFERENCES

R. Hill, E. H. Lee and S. J. Tupper, The theory of wedge indentation of ductile materials. Proc. R. Sot. A. 188, 273-289 (1947). D. S. Dugdale, Wedge indentation experiments with cold worked metals. J. Mech. Phys. Solids 2, 14-26 (1953). J. B. Pethica, R. Hutchings and W. C. Oliver, Hardness measurement at pentration depths as small as 20nm. Phil. Mug. A 48, 593-606 (1983). W. C. Oliver and C. J. McHargue, Characterising hardness and modulus of thin films-using a mechanical uronerties microprobe. Thin So/id Films 161, 117-122 i1988). _ A. G. Tangena and G. A. M. Hurkx, The determination of stress-strain curves of thin layers using indentation tests. Trans. ASME J. Engng Mater. Technol. 108, 230-232 (1986). D. Stone, An investigation of hardness and adhesion of sputter deposited Al on Si by a continuous indentation test. J. Muter. Res. 3, 141-147 (1988). C. H. Lee, S. Masaki and S. Kobayashi, Analysis of ball indentation. Int. J. mech. Sci. 14, 417 (1972). P. S. Follansbee and G. B. Sinclair, Quasi-static normal indentation of an elastoplastic half-space by a rigid sphere-I. Int. J. Solids Struct. 20, 81 (1984). K. Komvopoulos, Elasto-plastic finite element analysis of indented lavered media. J. Tribal. 111. 4306439 (1989). Z A. K. Bhattacharya and W. D. Niz, Finite element simulation of indentation experiments. Int. J. Solids Struct. 24, 881-891 (1988). J. 0. Hallquist, A numerical treatment of sliding interfaces and impact. In Computaiional Techniques for Interface Problems (Edited by K. C. Park and D. K. Garling), Vol. 30, pp. 117-133. ASME AMD (1978).

12. K. J. Bathe and A. Chaudhary, A solution method for planar and axisymmetric contact problems. Int. J. numer. Mesh. Engng 21, 65-88 (1985).


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P. Wriggers, T. Vu Van and E. Stein, Finite element formulations of large deformation impact-contact problems with friction. Comput. Strucr. 37, 319-331 (1990). R. L. Taylor and P. Papadopoulos, On a patch test for contact problems in two dimensions. In: Nonlinear Computational Mechanics. pp. 690-702, Springer, Berlin (1991). 0. C. Zienkiewicz, The Finite Element Method, 3rd Edn, Chap. 24 (Edited by R. L. Taylor). McGraw-Hill, London (1977). R. M. McMeeking and J. R. Rice. Finite element formulation for problems of large elastic-plastic deformation. Int. J. Solids Sfruct. 11, 601-616 (1975). J. C. Nagtegaal, D. M. Parks and J. R. Rice, On numerically accurate finite element solutions in the fully plastic range. Comput. Meth. appl. mech. Engng 4, 153-177 (1974). T. J. R. Hughes and J. Winget, Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis. Int. J. numer. Meth. Engng 15, 186221867 (1980). K. R. Jayadevan, Experimental and numerical studies of indentation mechanics of layered solids, M. E. disser- tation thesis, Dept of Mech. Engng, IISc, Bangalore, India ( 1994). R. D. Cook, D. S. Mahhus, and M. E. Plesha, Concepts and Application of Finite Element Analysis, 3rd Edn. Wiley, New York (1988). AM. Habraken and S. Cescotto, An automatic remesh- ing technique for finite element simulation of forming processes. Int. J. numer. Meth. Engng 30, 1503-1525 (1990). E. Hinton and J. S. Campbell. Local and global smoothing of discontinuous finite element functions using a least squares method. Int. J. numer. Merh. Engng 8, 461-480 (1974). S. Sebastian and S. K. Biswas, Effect of interface friction on the mechanics of indentation of a finite layer resting on a rigid substrate. J. Phys. D: appi. Phys. 24, 1131-1140 (1991).

APPENDIX A. LOCATION OF NEW GAUSS QUADRATURE POINTS IN OLD MESH

After re-zoning, each Gauss quadrature point of the elements in the re-zoned mesh has to be located within the distorted elements of the old mesh. Attention will be restricted to four-noded quadrilateral elements only. The concept behind isoparametric mapping is exploited to check whether a given point is inside or outside a given quadrilateral. After locating the quadrilateral element of the old mesh within which a particular Gauss quadrature point of the new mesh lies, its natural coordinates ([, 9) with respect to that quadrilateral element are determined by the same algorithm. The following steps are employed in the algorithm for each Gauss quadrature point G (see Fig. Al) of the re-zoned mesh.

(1) Each element A of the old mesh is first enclosed within a rectangle B as shown in Fig. Al.

(2) The (X, Y) co-ordinates of the Gauss quadrature point G which is considered are used to check whether it lies inside or outside the rectangle B.

(3a) If the Gauss quadrature point G lies outside the rectangle B, the algorithm returns to the step (1) and continues with the next element of the old mesh.

(3b) If G is inside rectangle B (see Fig. Al), control is passed to a separate module to examine whether it lies inside or outside the quadrilateral A. The steps followed in the above mentioned module are explained below.

1

Fig. Al. A schematic showing the distorted element which is enclosed in a rectangle.

A.1 Procedure for checking whether point G is inside quadrilateral A

The Gauss quadrature point G which lies inside the rectangle B is checked in this module, to examine whether that point lies inside or outside the quadrilateral A. The computational steps involved are presented below.

The natural coordinates (cc, qc) of the Gauss point G with respect to the element A in the old mesh (see Fig. Al) are computed using the inverse of the following isoparametric mapping equations:

-f: N,X,= X,, (Al) ,=I

c N,Y,= Y,. (-0 I- I

Here, N,([,, qc) is the parent shape function evaluated at (cc, qG), and X, and Y, are the Cartesian co-ordinates of the node I. Further, X, and Y, are the known Cartesian coordinates of the Gauss point G which is considered.

On substituting the shape functions and nodal coordinates of element A and also the Cartesian coordinates of the Gauss point G in the eqn (Al), the following set of simultaneous algebraic equations are obtained for the natural coordinates cc and qc:

(A3) h, f b,i, + b,rlc + b‘,icqG = 0.

Here a,, a,, and b,, b,, are constants which are given below.

a,=(X,+X2+X3+X-4XG)

a>=(-X,+X,+X,-XX,)

a,=(-X,-X2+X3+X4)

a4 = (X, - X, + X, - X,).

(A4)

The constants b,, b,, b,, 6, will be similar to the above expressions with Y, , Y,, Y,, Y4 and Yo used in place of X, , X,, X,, X, and X,, On expressing r7G in terms of cc from the eqn (A2), and substituting in eqn (A2). the following quadratic equation is obtained:

(a,b,-a,b,)1t-+(a,b,+a,b,

-a,b,-a,b,)&+(a,b,-a,b,)=O. (A5)


From the above equation, the two roots of cc are further consideration. Using the above selected value of cc, determined. The criterion which was adopted to check the other natural coordinate r~c is calculated using eqn (A2). whether the point G lies inside or outside the quadrilateral If this value of Rio is also within the limit (- I and + l), then A is that both the natural coordinates of any point within it is confirmed that the Gauss quadrature point G belongs or on the quadrilateral should lie in the range to element A. Otherwise, G is outside the quadrilateral A - 1 < (Y, q Q + 1. If one of the roots for & obtained from and the algorithm returns to step (I) and continues with the solving eqn (A4) satisfies this criterion, it is selected for next element.

finite element simulation of wedge indentation

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