• Open Access

An augmentation technique for large deformation frictional contact problems

Authors


Correspondence to: Hesch, Chair of Computational Mechanics, Department of Mechanical Engineering, University of Siegen, Germany.

E-mail: christian.hesch@uni-siegen.de

SUMMARY

The present work deals with a new approach to frictional large deformation contact problems. In particular, a new formulation of the frictional kinematics is introduced that is based on a specific augmentation technique used for the introduction of additional variables. This augmentation technique substantially simplifies the formulation of the whole system. A size reduction of the resulting system of algebraic equations is proposed. Consequently, the augmentation technique does not lead to an increase in size of the algebraic system of equations to be ultimately solved. The size reduction retains the simplicity of the formulation and preserves important conservation laws such as conservation of angular momentum. Copyright © 2013 John Wiley & Sons, Ltd.

1 INTRODUCTION

The development of numerical methods for frictional contact problems is one of the most challenging task in the field of computational sciences. It combines several fields of research, including optimization theory to deal with arising inequality constraints, the development of appropriate discretization schemes in space and time, and tribology for the physical modeling of the surface interaction.

The numerical model to be developed can be subdivided into different tasks, using constitutive laws to describe the physical nature of the surface interaction and, on the other hand, kinematic relations to embed these laws into the generally nonlinear finite element method. We omit any small deformation considerations and focus on the large deformation frictional contact problem. For a comprehensive survey of contemporary methods, we refer to the textbooks of Laursen [1] and Wriggers [2].

Within this paper, we focus on the development of a new method for the contact kinematics. The governing constitutive laws have been extensively investigated by several authors (see among others He and Curnier [3] and Laursen and Oancea [4]). We apply the well-known Coulomb's law, using the established analogy between Coulomb friction and nonassociative plasticity (see for example [5]). However, the proposed approach is capable to embed all kinds of constitutive friction laws.

Our approach follows the lines of the work of Laursen [1, 6], referred to herein as the direct approach [7]. Therefore, in contrast to earlier works (see for example Wriggers and Simo [8] and Parisch [9]), continuum mechanical arguments are used to derive the underlying variational formulation.

In contrast to previous works dealing with large deformation frictional contact, we propose a new augmentation technique for the description of the frictional kinematics. The augmentation technique relies on the introduction of additional variables representing the local convective coordinates. Note that the terminus augmentation is widely used for different methods, including the augmented Lagrangian update algorithm. Within our newly proposed method for frictional contact, we refer to augmentation techniques as used in the context of multibody systems [10]. The additional variables are connected to the original ones by introducing additional algebraic constraints that are enforced by means of Lagrange multipliers. When compared with the traditional direct approach, the augmentation technique significantly simplifies the whole formulation.

After the discretization in space has been performed, the resulting differential-algebraic equations can be reformulated to reduce the size of the algebraic system to be eventually solved. That is, applying a size-reduction procedure within the discrete setting essentially recovers the size of the original system. It is worth noting that the size-reduction process does not affect the algorithmic conservation properties of the proposed discretization in space and time. In particular, our discretization approach inherits the conservation laws for linear and angular momentum from the underlying continuous formulation.

The development of structure-preserving time-stepping schemes for large deformation contact problems has been subject of extensive research, see Laursen and Chawla [11, 12], Armero and Petöcz [13, 14], and Hesch and Betsch [15, 16]. Similar to these works, the spatial discretization of the contact surface used in the present work is based on the node-to-surface (NTS) method. The extension to more sophisticated models using mortar-based formulations [17-22] will be dealt with in a subsequent work.

The article is organized as follows. The fundamental equations of the underlying problem in strong and weak form are outlined in Section 2. In this connection, the well-established description of the frictional kinematics is summarized, and the newly proposed augmentation technique is introduced. The spatial discretization and the NTS method together with the specific formulation of the augmentation technique is given in Section 3. In Section 4, we apply a suitable time integration scheme and verify algorithmic conservation of linear and angular momentum. Representative numerical examples are presented in Section 5. Eventually, conclusions are drawn in Section 6.

2 GOVERNING EQUATIONS

We consider continuum bodies math formula written in their reference configurations for the large deformation problem at hand. To characterize the deformation, we assume the existence of a mapping math formula, math formula, such that we can introduce the deformation gradient math formula as follows:

display math(1)

Note that math formula remains nonsingular and invertible throughout the considered time interval math formula, that is, math formula. The material behavior is governed by the strain energy function math formula where math formula, math formula denotes the right Cauchy–Green tensor, and we define the first Piola–Kirchhoff stress tensor as follows:

display math(2)

Eventually, the balance of linear momentum reads

display math(3)

where math formula denotes the body force per reference volume and math formula the inertia term using the reference density math formula. The boundaries are subdivided into the Dirichlet boundary math formula, the Neumann boundary math formula, and the contact boundary math formula (Figure 1). We require that these boundaries do not overlap; hence, they satisfy

display math(4)

Moreover, appropriate boundary conditions are given by

display math(5)
display math(6)

where math formula denotes the unit outward normal. We assume that the bodies are in contact within the considered time interval math formula and restrict our consideration on a two body contact problem neglecting self-contact for simplicity of exposition. Additionally, we provide the initial conditions

display math(7)
display math(8)

finalizing the strong form of the problem. Next, we rewrite the system in weak form to obtain the virtual work of the whole system. To this end, we define the solution space

display math(9)

and the space of test functions

display math(10)

where the Sobolev space math formula consists of square-integrable functions and square-integrable first derivatives thereof. The virtual work contribution of each body reads

display math(11)

for math formula and math formula. Here, math formula denotes the second Piola–Kirchhoff stress tensor. Now, the principle of virtual work for the two-body contact problem under consideration can be written as math formula. Taking into account the balance of linear momentum across the contact interface

display math(12)

the contact contribution to the virtual work can be summarized in the expression

display math(13)

In the last statement, math formula contains the collection of the mappings math formula, i = 1,2 (similarly for math formula).

Figure 1.

Configurations of the two body contact problem (math formula: bodies in the reference (material) confi-guration, math formula: bodies in the current (spatial) configuration).

2.1 Contact formulation

We assume that a point math formula, math formula on the surface math formula is in contact with the opposing master surface math formula and define the projection

display math(14)

where math formula is the closest point to math formula. The master surface math formula itself can be viewed as a 2D manifold parametrized by the convective coordinates math formula, math formula. Thus, the projection can be characterized by the relationships

display math(15)

and

display math(16)

where math formula are calculated from (14). We further introduce tangent vectors

display math(17)

where math formula denotes the derivative with respect to math formula. Note that the vectors math formula are directed tangentially along the coordinate curves math formula at math formula (Figure 2). Associated dual vectors are defined by

display math(18)
Figure 2.

Parametrization of the master surface math formula.

where math formula is the inverse of the metric math formula. Next, we introduce the gap function

display math(19)

where math formula denotes the unit outward normal to math formula at math formula, defined as

display math(20)

Note that the tangent vectors math formula along with the normal vector math formula are covariant base vectors. Furthermore, the normal vector math formula is assumed to be directed tangentially along the coordinate line math formula (Figure 2). The variation of the gap function can now be written in the form

display math(21)

As usual, we decompose the contact traction in (13) into the normal and the tangential part

display math(22)

and require that math formula. For the normal component, the Karush–Kuhn–Tucker conditions

display math(23)

have to hold, whereas the vector math formula lies in the tangent space of the master surface math formula and can be resolved via

display math(24)

The corresponding frictional constitutive law to define the tractions math formula will be dealt with in the appendix.

Summarizing, the contact contribution to the virtual work can now be written in the form

display math(25)

The last statement depends crucially on the variation of the convective coordinates math formula on which we will focus next.

2.2 Frictional kinematics

Next, we focus on the variation of the convective coordinates to complete the contact formulation given in (25). In particular, we outline the most common approach, referred to as the direct approach in the following [23] and present subsequently a new augmentation technique for the description of the frictional kinematics.

Direct approach

The convective coordinates math formula can be obtained from the solution of the minimum distance problem (14). Correspondingly, the orthogonality condition

display math(26)

has to be valid. Computing the time derivative of the last equation yields

display math(27)

Using the unit length of the normal vector, that is, math formula together with math formula, we can rearrange the terms in (27) and obtain the rate of change of the convective coordinates

display math(28)

where math formula denotes the inverse of math formula and math formula is the curvature of the surface. Replacing the velocity by the variation yields

display math(29)

Assuming that math formula is valid at the contact interface, the variation of math formula boils down to

display math(30)

Accordingly, the virtual work expression (25) can be recast in the form

display math(31)

where relation (21) has been used. The majority of previous works dealing with large deformation frictional contact problems rely on (31) [2]. Note that statement (31) holds true if (29) is used instead of (30), because the additional terms to be considered only redefine the tractions math formula in tangential direction.

2.3 Coordinate augmentation technique

Following the arguments in Hesch & Betsch [16], we extend a specific coordinate augmentation technique to frictional contact problems. This technique relies on the introduction of additional coordinates math formula that represent the convective coordinates math formula. To link the new coordinates to the original ones, we introduce two constraint functions

display math(32)

and require that math formula. Similar to definition (17) for the tangent vectors, in (32), math formula for math formula. Analogous to the definition of the gap function (19), we introduce

display math(33)

where math formula again follows from (20) by replacing math formula with math formula. The contact contribution to the virtual work can now be determined along the lines of the direct approach. Accordingly, similar to (31), we obtained

display math(34)

where math formula. It is important to realize that the augmented coordinates math formula are to be viewed as primary variables on an equal footing with the original variables math formula. Consequently, the newly proposed augmentation technique strongly affects the discretization in space and time. Indeed, it will be shown in the sequel that the newly proposed augmentation technique simplifies the implementation significantly when compared with the direct approach.

3 SPATIAL DISCRETIZATION

We apply a suitable spatial discretization process to the bodies under consideration and to the contact constraints. More precisely, displacement-based finite elements are used for the bodies in contact, subdividing each body math formula into a finite number of elements math formula via

display math(35)

The polynomial approximations of the solution and the test space are written as

display math(36)

where math formula represents the nodal position at point math formula and math formula the corresponding variation at point math formula. Furthermore, math formula are global trilinear Lagrangian shape functions. The semi-discrete version of the principle of virtual work can now be written in the form

display math(37)

Note that the nodal position vectors math formula and math formula have been collected in the vector math formula (similarly for math formula). The last term in (37) denotes the discrete version of the contact virtual work, which will be dealt with in the following section. We can rewrite (37) as follows

display math(38)

where math formula represents the nodal mass contributions, math formula the internal nodal forces, and math formula the external forces. Throughout the paper, we assume that the internal and external forces are associated with a potential energy function

display math(39)

The last two terms in (38) represent the normal contact and frictional forces, which will be derived in the next section.

3.1 Node-to-surface element

Similar to the approximations of the solution and the test space, we define the following approximations at the contact boundaries

display math(40)

where math formula denote bilinear shape functions at the corresponding node math formula, representing the set of all nodes on the contact interface. Using the direct approach, we have to compute the convected coordinates math formula internally within each NTS element math formula (Figure 3) by solving

display math(41)

for the convective coordinates using a Newton–Raphson iteration. The discrete nodal gap function math formula (Figure 3) reads

display math(42)

using the set of nodes math formula. Employing the discrete nodal gap function, we can define the constraint function in normal direction

display math(43)

along with the Lagrange multipliers math formula that can be viewed as discrete counterpart of the normal traction math formula.

Figure 3.

Three-dimensional five node node-to-surface element.

The corresponding tangential tractions are dealt with in Appendix A for the case of Coulomb's law. Note that we incorporate the frictional response using a widely used regularization method based on a penalty parameter. Other types of enforcement of the frictional response are possible [24-26], because the proposed augmentation technique remains unaffected by, for example, the existence of additional Lagrange multipliers. The chosen approach is used for the sake of clarity in the presentation.

Similar to the kinematic relationship (29), the variation of the convective coordinates in the discrete setting reads

display math(44)

If we assume that the gap is zero, we obtain

display math(45)

and the corresponding discrete virtual contact work for a single NTS element reads

display math(46)

With regard to (38), we rearrange the frictional contributions using a single vector math formula. Furthermore, we collect all normal constraints in a single vector math formula and assemble the associated Lagrange multipliers in the vector math formula. Then, the semi-discrete equations of motion read

display math(47)

where math formula denotes the consistent mass matrix.

3.2 Coordinate augmentation technique

Next, we apply the coordinate augmentation technique described in Section 2.3 to the NTS element. In contrast to the direct approach, we calculate the convective coordinates on a global level, that is, we do not solve the algebraic system of equations in (41) internally but enforce them as additional constraints

display math(48)

Here, we make use of a vector math formula for each NTS element, representing the convective coordinates math formula. The associated Lagrange multipliers are given by math formula. In addition to that, the constraints in normal direction are given by

display math(49)

As before, we collect all data in global vectors, that is, we collect all augmented coordinates in a single vector math formula, where math formula denotes the number of all convective coordinates. Furthermore, the augmented constraints (48) are arranged in a single vector math formula and the associated Lagrange multipliers in a single vector math formula. The semi-discrete equations of motion can now be written as follows

display math(50)

where math formula and math formula. Furthermore, math formula math formula combines the frictional tractions in a single vector.

Implementation

To implement the newly proposed method in an efficient way, we eliminate the additional Lagrange multipliers math formula using the algebraic condition math formula. For a single NTS element, this condition reads

display math(51)

where math formula represents the tangential tractions of the corresponding NTS element. The Lagrange multipliers can now be calculated analytically as follows

display math(52)

Accordingly, on the level of each NTS element, the Lagrange multipliers associated with the augmented coordinates can be expressed in terms of the extended set of coordinates math formula and the contact traction math formula in normal direction. Using (52) for each NTS element, the vector math formula of Lagrange multipliers can be eliminated from the semi-discrete equations of motion (50). Accordingly, we arrive at

display math(53)

where the block diagonal matrix math formula consists of the local projection matrix

display math(54)

for each NTS element. Note that math formula is valid at the solution point, and we obtain the simplified system

display math(55)

The last set of equations defines the residual math formula, which we have to solve with respect to math formula, and math formula. This first reduction step can be written in matrix notation using the modified projection matrix

display math(56)

where math formula denotes the number of degrees of freedom of the configuration math formula, math formula the number of constraints math formula, and math formula the number of augmented coordinates math formula. Premultiplication of (50) by the projection matrix in (56) yields (55).

In a second step, we eliminate the augmented coordinates within the Newton–Raphson iteration

display math(57)

that is used to solve (55). Here, math formula and math formula denotes the derivative of math formula with respect to math formula and math formula, respectively. Next, we extract the equations for the augmented constraints of a single NTS element from (57)

display math(58)

and solve this last equation with respect to math formula, such that we obtain

display math(59)

Insertion in (57) yields the reduced system

display math(60)

The last reduction step can also be written in matrix notation using

display math(61)

It is important to remark that the whole reduction procedure can be carried out on element level for each single NTS-element, because math formula is block diagonal. The convective coordinates can be recovered using (59). The consistent linearization can now be carried out in two different ways:

  1. As shown in (57), we have to linearize math formula with respect to the configuration math formula and the augmented coordinates math formula. The involved constraints (48) and (49) are at most quadratic in the configuration and in the augmented coordinates, thus the only terms of higher order to be derived depending on the used constitutive law math formula (this derivative is always necessary) and the math formula inverse matrix math formula, that is, we have to linearize math formula.

  2. In math formula, we have used the projection matrix math formula to obtain a new residual, which we have to linearize to obtain the math formula matrix in (57). Alternatively, we can premultiply the full linearized original system (50) by math formula and obtain

    display math(62)

    where terms labeled by the upper index math formula represent the contributions arising from math formula and math formula. Next, we remove math formula and the corresponding columns from the system, because we solve directly for math formula using (52). The second reduction step follows as before, now avoiding the linearization of math formula. Note that we take again advantage of its block diagonal structure, such that all steps can be carried for each contact element.

The linearization is extremely simplified, compared with traditional methods, where we need to calculate the linearization of the variation of the convective coordinates (cf. Laursen [1])

display math(63)

where math formula has the same structure as math formula, given in (29).

Remark: Although we use Lagrange multipliers to enforce the normal constraints, we can also apply an augmented Lagrangian method to calculate the exact values of math formula.

3.3 Conservation properties

The conservation properties of the underlying mechanical system are well known, so we concentrate on the contact contributions. Reconsider the virtual work contributions of a single contact element

display math(64)

where we make use of (46) along with (42) and (44). The conservation of linear momentum may be verified by substituting math formula, where math formula is arbitrary and constant into the global virtual work of the constraint forces

display math(65)

using the direct approach in (44). To verify conservation of angular momentum, we substitute math formula and obtain

display math(66)

Note that the simplified variation (45) conserves angular momentum only if the normal gap is equal zero.

Finally, we verify the conservation properties of the augmented system in (50). The corresponding contact virtual work reads

display math(67)

where we have taken (math formula) math formula into account. Insertion of math formula into the augmented system yields

display math(68)

whereas insertion of math formula yields

display math(69)

Because the constraints are frame indifferent with respect to rigid body motions of the form

display math(70)

where math formula is a constant vector math formula, we can show that for each NTS element, the relation

display math(71)

is valid. Substituting math formula, math formula, math formula and subsequent derivation with respect to math formula yields

display math(72)

Thus, (68) holds for arbitrary math formula and linear momentum is conserved. Substituting math formula and math formula, where math formula is a skew-symmetric matrix, associated with the axial vector math formula so that math formula for any math formula, we end up with

display math(73)

Thus, (69) holds for arbitrary math formula and angular momentum is conserved for the semi-discrete system. The same statements are true for the reduced system, because the algebraic reformulation does not change the general characteristics of the system.

4 TEMPORAL DISCRETIZATION

The semi-discrete equations of motion (47) and (50) have to be discretized in time. Consider a time interval math formula subdivided into increments math formula and assume the state at time math formula to be known. Now, for a typical time step math formula, the full discrete version of (47) reads

display math(74)

Here, math formula denotes the discrete gradient of the strain energy function [27]. The discrete version of the frictional kinematics used in math formula is related to the definition of the convective coordinates

display math(75)

Note that we deal with the adjoint discrete traction math formula using a local evolution scheme in Appendix B. The time-discrete version of the augmented system in (50) reads

display math(76)

where math formula, math formula, and math formula. As already mentioned, math formula consists of the tractions math formula, see (50). Accordingly, math formula has to be evaluated as shown in Appendix B. Following the arguments outlined in the previous section, we create a local projection matrix as follows

display math(77)

and obtain for the reduced system

display math(78)

The second reduction step follows immediately from (60) using the discretized projection matrix in (77) evaluated at time math formula

display math(79)

The full discrete system to be solved in each Newton–Raphson iteration now reads

display math(80)

where math formula consists of the residual contributions in math formula, and math formula and math formula denotes the consistent linearization of math formula with respect to math formula and math formula, respectively. In a final step, we recover the augmented coordinates by solving

display math(81)

for each NTS element.

It is obvious that the linearization is extremely simplified compared with traditional schemes. Furthermore, the proposed scheme is more consistent because it ensures the exact fulfillment of the orthogonality conditions math formula at each time node within the chosen mid-point type scheme.

4.1 Conservation properties

As before, we focus on the contact contribution and begin with the verification of the conservation of linear momentum. To this end, we substitute math formula, math formula into the weak form of the contact contributions

display math(82)

which confirms that the constraints do not affect linear momentum conservation. Following the arguments in (66), we substitute math formula and obtain

display math(83)

which confirms that the constraints do not affect angular momentum conservation as well.

At last, we verify the conservation properties of the full discrete system in (76) and substitute math formula

display math(84)

whereas we obtain

display math(85)

if we substitute math formula. Once again frame indifference of the vector of constraints math formula against rigid body motions is crucial for the fulfillment of the conservation laws. Proceeding along the lines of Section 3.3, we can easily verify that

display math(86)

where math formula. Substituting math formula, math formula, math formula yields

display math(87)

Analogous to the semi-discrete system, linear momentum is algorithmically conserved. Substituting math formula and math formula yields

display math(88)

Thus, angular momentum is algorithmically conserved for the full-discrete system. Note that the last statement is also true for the reduced system, because the algebraic reformulation of the system does not change any properties of the underlying formulation.

5 NUMERICAL RESULTS

In this section, we evaluate the accuracy and performance of the newly proposed method and compare the results to well-known and established methods. To solve the arising nonlinear system of equations, a Newton–Raphson solution procedure has been implemented.

5.1 Contact of two elements

To evaluate the properties of the algorithms under consideration, we investigate a simple nonlinear three-dimensional example, which is constructed such that we obtain reproducible results. In particular, we consider two 3D elements, using trilinear shape functions. A compressible Neo–Hookean material is used with associated strain energy density function

display math(89)

where math formula and math formula, math formula are Lamé parameters corresponding to a Young's modulus of math formula and a Poisson's ratio of math formula. The reference density is given by math formula and the coefficient of friction math formula. The initial position of the 16 nodes are given in Table 1 together with the initial velocity (Figure 4). Because of the initial configuration, the tangent vectors math formula of the master surface are not orthonormal.

Figure 4.

Reference configuration and initial velocity math formula.

Table 1. Nodal positions and initial velocity.
NodePositionVelocityNodePositionVelocity
1[ − 0.5, − 1, 2.1][0, 0.1, − 0.04]1[ − 1, − 1, 1][0 0 0]
2[ − 0.5, 0, 2.1][0, 0.1, − 0.04]2[ − 1.5, 1.5, 1][0 0 0]
3[ − 0.5, − 1, 1.1][0, 0.1, − 0.04]3[ − 1, − 1, 0][0 0 0]
4[ − 0.5, 0, 1.1][0, 0.1, − 0.04]4[ − 1.5, 1.5, 0][0 0 0]
5[ 0.5, − 1, 2.1][0, 0.1, − 0.04]5[1, − 1, 1][0 0 0]
6[ 0.5, 0, 2.1][0, 0.1, − 0.04]6[1.2, 1, 1][0 0 0]
7[ 0.5, − 1, 1.1][0, 0.1, − 0.04]7[1, − 1, 0][0 0 0]
8[ 0.5, 0, 1.1][0, 0.1, − 0.04]8[1.2, 1, 0][0 0 0]

Both elements are flying free in space, that is, no boundary conditions are prescribed. First, we show results using an implicit Euler backward algorithm. Therefore, the z-position of node 8 of the upper block, which is directly in contact with the lower block, is plotted over time in Figures 5 and 6. In particular, Figure 5 shows the results for different time step sizes of the newly proposed algorithm using both reduction steps, whereas Figure 6 shows the results of the conventional direct approach. As expected, the Euler backward algorithm damps oscillations for larger time step sizes. In Figure 7, we show a comparison of the augmented system, the reduced system, and the direct approach for a time step size of math formula. The results coincide extremely well for the used implicit Euler backward algorithm. Thus, the advantages of the new approach considered here relies on the simplified structure of the contact element.

Figure 5.

Augmented coordinates, z-position of node 8 plotted over time.

Figure 6.

Direct approach, z-position of node 8 plotted over time.

Figure 7.

Comparison of the different approaches under consideration.

Figure 8 shows additionally the results of the proposed new algorithm using a mid-point type evaluation, as proposed in Section 4. Using the mid-point type evaluation, we obtain even for large time step sizes reliable results, for example, for math formula, using only 100 time steps for the whole simulation. Note that Lagrange multipliers have been used to enforce the constraints in normal direction throughout all shown examples.

Figure 8.

Augmented coordinates, mid-point type evaluation.

Finally, total energy and total angular momentum are plotted in Figures 9 and 10 for the proposed scheme using the mid-point type evaluation. As can be seen, total energy is conserved after the frictional impact. Furthermore, total angular momentum is also conserved. Although not shown here, linear momentum is also conserved algorithmically.

Figure 9.

Total energy over time.

Figure 10.

Total angular momentum over time.

5.2 Two tori impact problem

In this example, we consider an impact problem of two tori to demonstrate that the proposed algorithm is suitable for large systems. Initial values and the material properties have been taken from Yang and Laursen [28]. The initial configuration is displayed in Figure 11. The inner and outer radius of the tori are 52 and 100; the wall thickness of each hollow torus is 4.5. Both tori are subdivided into 8024 elements, using a Neo–Hookean hyperelastic material with math formula and math formula. The initial densities are math formula and the homogeneous, initial velocity of the left torus is given by math formula. A time step size of 0.01 has been used throughout the whole simulation. The deformation at different time steps is shown in Figure 12.

Figure 11.

Initial configuration of the two tori impact problem.

Figure 12.

Deformation at time 2.5 and 5.

The evolution of the total energy is shown in Figure 13, whereas the three components of angular momentum are shown in Figure 14. As expected, total energy decreases because of the frictional behavior. Because we used the proposed mid-point type evaluation of the system, angular momentum is conserved.

Figure 13.

Total energy plotted over time.

Figure 14.

Components of angular momentum plotted over time.

Finally, Figure 15 shows the deformation at math formula for different friction coefficients, using math formula and math formula. The deformation changes significantly because large sliding effects are directly correlated with the friction coefficient.

Figure 15.

Comparison at math formula for math formula (left) and math formula (right).

6 CONCLUSIONS

A novel formulation for the frictional kinematics has been developed in the framework of large deformation contact. This new method rests on an augmentation technique, which substantially simplifies the underlying expressions at the cost of an enlarged global system of algebraic equations to be solved. To remedy this drawback, a size-reduction procedure has been proposed on the basis of the elimination of the Lagrange multipliers associated with the augmented coordinates. In a second step, the size of the system has been further reduced to the original size. In this connection, a new analytical representation of the projection of the convective coordinates to the configuration space has been established.

The new approach is much more simple compared with traditional methods. As shown, the internal Newton iteration to determine the actual convective coordinates has been removed, and the residual and the tangent contributions of the contact element are significantly simplified. We have demonstrated the usability of the proposed augmentation technique for large deformation problems. Because the size-reduction of the system relies on an analytical reformulation, the underlying conservation laws are not affected. Thus, we could ensure algorithmic conservation of linear and angular momentum. This provides enhanced numerical stability of the method for large simulations.

APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS

Many researchers have investigated various constitutive laws used to describe the tangential tractions. We omit a further investigation and focus on a standard dry friction Coulomb law to complete the set of equations used for the numerical examples. On the basis of this specific formulation, we state that

display math(A.1)

The tangential displacement in the case of slip follows from math formula where math formula denotes the consistency parameter, which depends on (A.1), and we can write

display math(A.2)

In analogy to plasticity, we rewrite the last statement as follows

display math(A.3)

and regularize the equation for the tangential velocity math formula math formula using a penalty method

display math(A.4)

(Figure A.1). Note that the components in tangential direction read

display math(A.5)

where we have made use of math formula.

Figure A.1.

Admissible region for tangential traction math formula in case of Coulomb law.

APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS

Here, we apply a one-step integration scheme of the local evolution equations (A.5) following the arguments in the work of Armero and Petöcz [14]. Consequently, the approximation of the tractions can be written as follows

display math(B.1)

where math formula controls the corresponding time stepping scheme and should be chosen consistent with the global time stepping scheme. Taking the inequality conditions (A.3) into account, we obtain

display math(B.2)

Note that math formula is represented by a Lagrange multiplier, constant within the time step. To implement (B.2), we apply a return mapping scheme and start by considering the stick case, that is, math formula

display math(B.3)

which defines our trial state. Depending on the condition math formula, slip occurs, and we obtain

display math(B.4)

by comparing (B.2) math formula and (B.3) math formula. After short calculations using the relation math formula, the consistency parameter in the case of slip is determined by

display math(B.5)

Thus, the final contribution in the case of slip reads

display math(B.6)

To summarize, the return mapping scheme can be written as follows

display math(B.7)

which completes the used definition for the tractions.

ACKNOWLEDGEMENTS

Support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG) under grant HE 5943/1-1 and HE 5943/3-1. This support is gratefully acknowledged.

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