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Keywords:

  • frictional contact;
  • Coulomb's law;
  • node-to-surface method;
  • Lagrange multipliers;
  • augmentation

SUMMARY

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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 [6, 1], 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

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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

  • display math(1)

Note that inline image remains nonsingular and invertible throughout the considered time interval inline image, that is, inline image. The material behavior is governed by the strain energy function inline image where inline image, inline image 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 inline image denotes the body force per reference volume and inline image the inertia term using the reference density inline image. The boundaries are subdivided into the Dirichlet boundary inline image, the Neumann boundary inline image, and the contact boundary inline image (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 inline image denotes the unit outward normal. We assume that the bodies are in contact within the considered time interval inline image 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 inline image consists of square-integrable functions and square-integrable first derivatives thereof. The virtual work contribution of each body reads

  • display math(11)

for inline image and inline image. Here, inline image 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 inline image. 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, inline image contains the collection of the mappings inline image, i = 1,2 (similarly for inline image).

image

Figure 1. Configurations of the two body contact problem (inline image: bodies in the reference (material) confi-guration, inline image: bodies in the current (spatial) configuration).

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2.1 Contact formulation

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

  • display math(14)

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

  • display math(15)

and

  • display math(16)

where inline image are calculated from (14). We further introduce tangent vectors

  • display math(17)

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

  • display math(18)
image

Figure 2. Parametrization of the master surface inline image.

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where inline image is the inverse of the metric inline image. Next, we introduce the gap function

  • display math(19)

where inline image denotes the unit outward normal to inline image at inline image, defined as

  • display math(20)

Note that the tangent vectors inline image along with the normal vector inline image are covariant base vectors. Furthermore, the normal vector inline image is assumed to be directed tangentially along the coordinate line inline image (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 inline image. For the normal component, the Karush–Kuhn–Tucker conditions

  • display math(23)

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

  • display math(24)

The corresponding frictional constitutive law to define the tractions inline image 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 inline image 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 inline image 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, inline image together with inline image, we can rearrange the terms in (27) and obtain the rate of change of the convective coordinates

  • display math(28)

where inline image denotes the inverse of inline image and inline image is the curvature of the surface. Replacing the velocity by the variation yields

  • display math(29)

Assuming that inline image is valid at the contact interface, the variation of inline image 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 inline image 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 inline image that represent the convective coordinates inline image. To link the new coordinates to the original ones, we introduce two constraint functions

  • display math(32)

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

  • display math(33)

where inline image again follows from (20) by replacing inline image with inline image. 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 inline image. It is important to realize that the augmented coordinates inline image are to be viewed as primary variables on an equal footing with the original variables inline image. 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

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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 inline image into a finite number of elements inline image via

  • display math(35)

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

  • display math(36)

where inline image represents the nodal position at point inline image and inline image the corresponding variation at point inline image. Furthermore, inline image 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 inline image and inline image have been collected in the vector inline image (similarly for inline image). 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 inline image represents the nodal mass contributions, inline image the internal nodal forces, and inline image 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 inline image denote bilinear shape functions at the corresponding node inline image, representing the set of all nodes on the contact interface. Using the direct approach, we have to compute the convected coordinates inline image internally within each NTS element inline image (Figure 3) by solving

  • display math(41)

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

  • display math(42)

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

  • display math(43)

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

image

Figure 3. Three-dimensional five node node-to-surface element.

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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 inline image. Furthermore, we collect all normal constraints in a single vector inline image and assemble the associated Lagrange multipliers in the vector inline image. Then, the semi-discrete equations of motion read

  • display math(47)

where inline image 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 inline image for each NTS element, representing the convective coordinates inline image. The associated Lagrange multipliers are given by inline image. 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 inline image, where inline image denotes the number of all convective coordinates. Furthermore, the augmented constraints (48) are arranged in a single vector inline image and the associated Lagrange multipliers in a single vector inline image. The semi-discrete equations of motion can now be written as follows

  • display math(50)

where inline image and inline image. Furthermore, inline image inline image 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 inline image using the algebraic condition inline image. For a single NTS element, this condition reads

  • display math(51)

where inline image 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 inline image and the contact traction inline image in normal direction. Using (52) for each NTS element, the vector inline image 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 inline image consists of the local projection matrix

  • display math(54)

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

  • display math(55)

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

  • display math(56)

where inline image denotes the number of degrees of freedom of the configuration inline image, inline image the number of constraints inline image, and inline image the number of augmented coordinates inline image. 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, inline image and inline image denotes the derivative of inline image with respect to inline image and inline image, 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 inline image, 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 inline image 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 inline image with respect to the configuration inline image and the augmented coordinates inline image. 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 inline image (this derivative is always necessary) and the inline image inverse matrix inline image, that is, we have to linearize inline image.

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

    • display math(62)

    where terms labeled by the upper index inline image represent the contributions arising from inline image and inline image. Next, we remove inline image and the corresponding columns from the system, because we solve directly for inline image using (52). The second reduction step follows as before, now avoiding the linearization of inline image. 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 inline image has the same structure as inline image, 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 inline image.

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 inline image, where inline image 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 inline image 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 (inline image) inline image into account. Insertion of inline image into the augmented system yields

  • display math(68)

whereas insertion of inline image yields

  • display math(69)

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

  • display math(70)

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

  • display math(71)

is valid. Substituting inline image, inline image, inline image and subsequent derivation with respect to inline image yields

  • display math(72)

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

  • display math(73)

Thus, (69) holds for arbitrary inline image 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

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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

  • display math(74)

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

  • display math(75)

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

  • display math(76)

where inline image, inline image, and inline image. As already mentioned, inline image consists of the tractions inline image, see (50). Accordingly, inline image 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 inline image

  • display math(79)

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

  • display math(80)

where inline image consists of the residual contributions in inline image, and inline image and inline image denotes the consistent linearization of inline image with respect to inline image and inline image, 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 inline image 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 inline image, inline image 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 inline image 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 inline image

  • display math(84)

whereas we obtain

  • display math(85)

if we substitute inline image. Once again frame indifference of the vector of constraints inline image 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 inline image. Substituting inline image, inline image, inline image yields

  • display math(87)

Analogous to the semi-discrete system, linear momentum is algorithmically conserved. Substituting inline image and inline image 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

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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 inline image and inline image, inline image are Lamé parameters corresponding to a Young's modulus of inline image and a Poisson's ratio of inline image. The reference density is given by inline image and the coefficient of friction inline image. 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 inline image of the master surface are not orthonormal.

image

Figure 4. Reference configuration and initial velocity inline image.

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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 inline image. 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.

image

Figure 5. Augmented coordinates, z-position of node 8 plotted over time.

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image

Figure 6. Direct approach, z-position of node 8 plotted over time.

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image

Figure 7. Comparison of the different approaches under consideration.

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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 inline image, 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.

image

Figure 8. Augmented coordinates, mid-point type evaluation.

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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.

image

Figure 9. Total energy over time.

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image

Figure 10. Total angular momentum over time.

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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 inline image and inline image. The initial densities are inline image and the homogeneous, initial velocity of the left torus is given by inline image. 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.

image

Figure 11. Initial configuration of the two tori impact problem.

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image

Figure 12. Deformation at time 2.5 and 5.

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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.

image

Figure 13. Total energy plotted over time.

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image

Figure 14. Components of angular momentum plotted over time.

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

image

Figure 15. Comparison at inline image for inline image (left) and inline image (right).

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6 CONCLUSIONS

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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 inline image where inline image 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 inline image inline image 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 inline image.

image

Figure A.1. Admissible region for tangential traction inline image in case of Coulomb law.

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

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES

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 inline image 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 inline image 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, inline image

  • display math(B.3)

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

  • display math(B.4)

by comparing (B.2) inline image and (B.3) inline image. After short calculations using the relation inline image, 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.

REFERENCES

  1. Top of page
  2. SUMMARY
  3. 1 INTRODUCTION
  4. 2 GOVERNING EQUATIONS
  5. 3 SPATIAL DISCRETIZATION
  6. 4 TEMPORAL DISCRETIZATION
  7. 5 NUMERICAL RESULTS
  8. 6 CONCLUSIONS
  9. APPENDIX A: CONSTITUTIVE EVOLUTION EQUATIONS
  10. APPENDIX B: LOCAL TIME STEPPING SCHEME FOR THE FRICTIONAL EVOLUTION EQUATIONS
  11. ACKNOWLEDGEMENTS
  12. REFERENCES
  • 1
    Laursen TA.Computational Contact and Impact Mechanics. Springer-Verlag: Berlin, Heidelberg, 2002.
  • 2
    Wriggers P.Computational Contact Mechanics, 2nd edition. Springer-Verlag: New York, 2006.
  • 3
    He QC, Curnier A.Anisotropic dry friction between two orthotropic surfaces undergoing large displacements. European Journal of Mechanics - A/Solids 1993; 12(5):631666.
  • 4
    Laursen TA, Oancea VG.On the constitutive modeling and finite element computation of rate-dependent frictional sliding in large deformations. Computer Methods in Applied Mechanics and Engineering 1997; 143:197227.
  • 5
    Simo JC, Hughes TJR.Computational inelasticity. Springer-Verlag: New York, Berlin, 1997.
  • 6
    Laursen TA, Simo JC.A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems. International Journal for Numerical Methods in Engineering 1993; 36:34513485.
  • 7
    Konyukhov A.Geometrically Exact Theory for Contact Interactions. Habilitationsschrift edition. KIT: Karlsruhe, 2010.
  • 8
    Wriggers P, Simo J.A note on tangent stiffnesses for fully nonlinear contact problems. Communications in Numerical Methods in Engineering 1985; 1:199203.
  • 9
    Parisch H.A consistent tangent stiffness matrix for three-dimensional non-linear contact analysis. International Journal for Numerical Methods in Engineering 1989; 28:19031812.
  • 10
    Betsch P, Uhlar S.Energy-momentum conserving integration of multibody dynamics. Multibody System Dynamics 2007; 17(4):243289.
  • 11
    Laursen TA, Chawla V.Design of energy conserving algorithms for frictionless dynamic contact problems. International Journal for Numerical Methods in Engineering 1997; 40:863886.
  • 12
    Chawla V, Laursen TA.Energy consistent algorithms for frictional contact problems. International Journal for Numerical Methods in Engineering 1998; 42:799827.
  • 13
    Armero F, Petöcz E.Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems. Computer Methods in Applied Mechanics and Engineering 1998; 158:269300.
  • 14
    Armero F, Petöcz E.A new dissipative time-stepping algorithm for frictional contact problems: Formulation and analysis. Computer Methods in Applied Mechanics and Engineering 1999; 179:159178.
  • 15
    Hesch C, Betsch P.Transient 3D Domain Decomposition Problems: Frame-indifferent mortar constraints and conserving integration. International Journal for Numerical Methods in Engineering 2010; 82:329358.
  • 16
    Hesch C, Betsch P.Transient 3D contact problems – NTS method: Mixed methods and conserving integration. Computational Mechanics 2011; 48:437449.
  • 17
    McDevitt TW, Laursen TA.A mortar-finite element formulation for frictional contact problems. International Journal for Numerical Methods in Engineering 2000; 48:15251547.
  • 18
    Hüeber S, Wohlmuth BI.A primal-dual active set strategy for non-linear multibody contact problems. Computer Methods in Applied Mechanics and Engineering 2005; 194:31473166.
  • 19
    Hager C, Hüeber S, Wohlmuth B.A stable energy conserving approach for frictional contact problems based on quadrature formulas. International Journal for Numerical Methods in Engineering 2008; 73:205225.
  • 20
    Hesch C, Betsch P.A comparison of computational methods for large deformation contact problems of flexible bodies. ZAMM 2006; 86:818827.
  • 21
    Hesch C, Betsch P.A mortar method for energy-momentum conserving schemes in frictionless dynamic contact problems. International Journal for Numerical Methods in Engineering 2009; 77:14681500.
  • 22
    Hesch C, Betsch P.Transient 3D contact problems – Mortar method: Mixed methods and conserving integration. Computational Mechanics 2011; 48:461475.
  • 23
    Konyukhov A, Schweizerhof K.Covariant description for frictional contact problems. Computer Methods in Applied Mechanics and Engineering 2005; 35:190213.
  • 24
    Gitterle M, Popp A, Gee W, Wall WA.Finite deformation frictional mortar contact using a semi-smooth Newton method with consistent linearization. International Journal for Numerical Methods in Engineering 2010; 84:543571.
  • 25
    Hüeber S, Stadler G, Wohlmuth BI.A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction. SIAM Journal on Scientific Computing 2008; 30:572596.
  • 26
    Tur M, Fuenmayor FJ, Wriggers P.A mortar-based frictional contact formulation for large deformations using Lagrange multipliers. Computer Methods in Applied Mechanics and Engineering 2009; 198:28602873.
  • 27
    Betsch P, Steinmann P.Conservation Properties of a Time FE Method. Part III: Mechanical systems with holonomic constraints. International Journal for Numerical Methods in Engineering 2002; 53:22712304.
  • 28
    Yang B, Laursen TA.A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulation. Computational Mechanics 2008; 41:189205.