A computational approach for thermoelastoplastic frictional contact based on a monolithic formulation using nonsmooth nonlinear complementarity functions
 Alexander Seitz^{1}Email authorView ORCID ID profile,
 Wolfgang A. Wall^{1} and
 Alexander Popp^{2}
https://doi.org/10.1186/s4032301800983
© The Author(s) 2018
Received: 12 April 2017
Accepted: 14 February 2018
Published: 2 March 2018
Abstract
A new monolithic solution scheme for thermoelastoplasticity and thermoelastoplastic frictional contact with finite deformations and finite strains is presented. A key feature is the reformulation of all involved inequality constraints, namely those of Hill’s orthotropic yield criterion as well as the normal and tangential contact constraints, in terms of nonsmooth nonlinear complementarity functions. Using a consistent linearization, this system of equations can be solved with a nonsmooth variant of Newton’s method. A quadrature pointwise decoupled plastic constraint enforcement and the use of socalled dual basis functions in the mortar contact formulation allow for a condensation of all additionally introduced variables, thus resulting in an efficient formulation that contains discrete displacement and temperature degrees of freedom only, while, at the same time, an exact constraint enforcement is assured. Numerical examples from thermoplasticity, thermoelastic frictional contact and thermoelastoplastic frictional contact demonstrate the wide range of applications covered by the presented method.
Keywords
Introduction
In many engineering applications frictional contact and elastoplastic material behavior come hand in hand. Just one class of typical wellknown examples are metal forming and impact/crash analysis, where, at high strain rates, thermal effects need to be taken into account. The thermomechanical coupling appears in several forms: firstly and most obviously, there is heat conduction across the contact interface. Secondly, the dissipation of frictional work leads to an additional heating at the contact interface. Thirdly, also plastic work within the structure is transformed to heat. Vice versa, the current temperature may influence the elastic and especially the plastic material response. All this necessitates robust and efficient solution algorithms for fully coupled thermoelastoplastic contact problems, which has been an active research topic over the past 25 years. Most contributions, however, focus either on thermoplasticity or on thermomechanical contact, while resorting to relatively simple standard methods for the remaining problem parts.
Early implementations of thermoelastic contact based on nodetosegment contact formulations in combination with a penalty constraint enforcement can be found in [1–7]. Within the last decade, more sophisticated variationally consistent contact discretizations based on the mortar method have been developed and applied to thermomechanical contact in [8–12]. In addition, those algorithms satisfy the contact constraints exactly (at least in a weak sense) by using either Lagrange multipliers or an augmented Lagrangian functional instead of a simple penalty approach. Due to an easier implementation and other benefits like symmetric operators, most of the cited works above employ some sort of partitioned solution scheme for solving the structural problem (at constant temperature) and thermal problem (at constant displacement) sequentially. In thermoplasticity, those partitioned schemes based on an isothermal split are only conditionally stable [13]. Only [4, 6, 11, 12] employ monolithic solution schemes, which solve for displacements and temperatures simultaneously. Most developments of advanced computational methods in thermomechanical contact are restricted to thermoelastic effects; coupled thermoelastoplastic contact can only be found in [5–7].
Numerical algorithms for finite deformation thermoplasticity go back to the seminal work by Simo and Miehe [13], which is based on the isothermal radial return mapping algorithm presented in [14, 15]. Both partitioned and monolithic solution approaches are discussed in [13]. Several extensions to this algorithm have been presented later, e.g. a monolithic formulation in principle axes [16] and a variant including temperaturedependent elastic material properties [17]. In a different line of work, a variational formulation of thermoplasticity has been developed in [18], where the rate of plastic work converted to heat follows from a variational principle instead of being a (constant) material parameter as in [13]. A comparison to experimental results is presented in [19] to support this variational form. We point out that both approaches to determine the plastic dissipation, i.e. [13] and [18], are applicable within the algorithm for thermoplasticity that will be derived in this manuscript. Besides the mentioned radial return mapping and variational formulations, a different numerical algorithm to isothermal plasticity at finite strains has been developed in [20]. Based on fundamental ideas from [21], the plastic deformation at every quadrature point is introduced as an additional primary variable and the plastic inequality constraints are reformulated as nonlinear complementarity functions. This allows for a constraint violation during the nonlinear solution procedure, i.e. in the preasymptotic range of Newton’s method, while ensuring their satisfaction at convergence. As usual in computational plasticity, the material constraints are enforced at each material point independently, such that the additional unknowns can be condensed directly at quadrature point level. It could be shown in [20] that due to this less restrictive formulation, a higher robustness can be achieved, which allows for larger time or load steps.
The present paper now aims at developing a monolithic solution scheme for the thermoelastoplastic frictional contact problem based on a new approach. Mortar finite element methods with dual Lagrange multipliers are applied for the contact treatment using nonlinear complementarity functions to deal with both the inequality constraints arising from frictional contact as well as plasticity in a unified manner. This bears novelty both for the numerical formulation of anisotropic thermoplasticity within the bulk material as well as for the fully nonlinear thermomechanical contact formulation at the interface. Furthermore, full compatibility of the algorithms for thermoplasticity and thermomechanical contact is demonstrated. Concerning plasticity, an extension of [20] to coupled thermoplasticity within a monolithic solution framework is presented. Similar to the isothermal case, the use of Gausspointwise decoupled plastic deformation allows for a condensation of the additionally introduced plastic unknowns, where now also thermomechanical coupling terms have to be accounted for. The novel thermomechanical contact formulation represents a fully nonlinear extension of [12] including a consistent linearization with respect to both the displacement and temperature unknowns. Moreover, the use of dual Lagrange multipliers within a mortar contact formulation enables the trivial condensation of the discrete contact Lagrange multipliers such that the final linearized system to be solved consists of displacement and temperature degrees of freedom only. Our new thermomechanical contact formulation is applicable for both classical finite elements based on Lagrange polynomial basis functions as well as isogeometric analysis using NURBS basis functions, for which an appropriate dual basis has recently been proposed in [22]. Owing to the variational basis of the mortar method, the thermomechanical contact patch test on nonmatching discretizations is satisfied exactly and optimal convergence rates are achieved. Since this paper touches on various topics, and not every reader may be familiar with every single topic, we try to give a selfcontained and rather detailed description of the different subproblems and solution approaches. Even though this requires to some extent the repetition of methods developed elsewhere, we hope to thereby make the article and its novelties amenable to a broader audience.
The remainder of this paper is outlined as follows: “Thermoplasticity in the bulk continuum” section contains the treatment of thermoplasticity within the bulk structure from the underlying continuum mechanics to the final discrete system. “Thermomechanical contact” section then incorporates thermomechanical contact, again starting from a continuum description and closing with the linearized system that needs to be solved in each Newton iteration step. Finally, several challenging numerical examples in “Numerical results” section demonstrate the high accuracy and robustness that can be achieved for benchmark tests and more complex applications in thermoelastoplasticity, thermoelastic contact and thermoelastoplastic contact.
Thermoplasticity in the bulk continuum
Before introducing the new algorithmic treatment of thermoplasticity in “Solution algorithm using nonlinear complementarity functions” section, we summarize the wellknown constitutive relations of thermomechanics in “Continuum thermomechanics and thermoplasticity” and “Yield criterion and evolution of internal variables” sections, for which more details can be found in the literature, see e.g. [13, 23, 24]. Only for the simplicity of presentation and not due to any particular restrictions of the developed framework, a few assumptions commonly used in e.g. [13, 17] are adopted here and outlined in “Assumptions on the used free energy” section. In “Weak form of the thermomechanical problem” and “Spatial discretization of the thermomechanical continuum” sections and, the finite element discretization of the problem (see e.g. [24, 25] and many others) is derived, whereas “Time discretization” section outlines a time discretization based on [26, 27]. Finally, in “Solution algorithm using nonlinear complementarity functions” section, those developments are used to construct a novel nonlinear solution procedure for thermoplasticity using nonlinear complementarity functions. This can be seen as an extension of the authors’ previous work [20], where increased robustness in the isothermal case as compared to classical return mapping algorithms had been demonstrated in the isothermal case.
Continuum thermomechanics and thermoplasticity

Balance of mass:$$\begin{aligned} \dot{\rho }_0&= 0 \quad \text { in }\Omega \times (0,t_E], \end{aligned}$$(1)

Balance of linear momentum:$$\begin{aligned} \mathrm {Div}{{\mathbf {P}}}+\hat{{{\mathbf {b}}}}_0&= \rho _0 \ddot{{{\mathbf {u}}}}\;\; \quad \text { in }\Omega \times (0,t_E], \end{aligned}$$(2)

Balance of angular momentum:$$\begin{aligned} {{\mathbf {P}}}{{\mathbf {F}}}^\mathrm {T}&= {{\mathbf {F}}}{{\mathbf {P}}}^\mathrm {T} \quad \text { in }\Omega \times (0,t_E], \end{aligned}$$(3)

Balance of energy:$$\begin{aligned} \dot{E}  {{\mathbf {P}}}:\dot{{{\mathbf {F}}}} + \mathrm {Div}{{\mathbf {Q}}}  R&= 0 \quad \text { in }\Omega \times (0,t_E], \end{aligned}$$(4)

Clausius–Duhem inequality:$$\begin{aligned} {{\mathbf {P}}}:\dot{{{\mathbf {F}}}}  \dot{E} + T\dot{\eta }  \frac{1}{T}{{\mathbf {Q}}}\cdot \mathrm {Grad}T&\ge 0 \quad \text { in }\Omega \times (0,t_E], \end{aligned}$$(5)

Displacement Dirichlet boundary condition:$$\begin{aligned} {{\mathbf {u}}}&=\hat{{{\mathbf {u}}}} \quad \text { on }\Gamma _u^D\times (0,t_E], \end{aligned}$$(6)

Displacement Neumann boundary condition:$$\begin{aligned} {{\mathbf {P}}}{{\mathbf {N}}}&=\hat{{{\mathbf {t}}}}_0 \quad \text { on }\Gamma _u^N\times (0,t_E], \end{aligned}$$(7)

Temperature Dirichlet boundary condition:$$\begin{aligned} T&=\hat{T} \quad \text { on }\Gamma _T^D\times (0,t_E], \end{aligned}$$(8)

Temperature Neumann boundary condition:$$\begin{aligned} {{\mathbf {Q}}}{{\mathbf {N}}}&=\hat{Q}_0 \quad \text { on }\Gamma _T^N\times (0,t_E], \end{aligned}$$(9)

Initial displacement:$$\begin{aligned} {{\mathbf {u}}}&={{\mathbf {u}}}_0 \quad \text { in }\Omega \times 0 , \end{aligned}$$(10)

Initial velocity:$$\begin{aligned} \dot{{{\mathbf {u}}}}&=\dot{{{\mathbf {u}}}}_0 \quad \text { in }\Omega \times 0, \end{aligned}$$(11)

Initial temperature:$$\begin{aligned} T&=T_0 \quad \text { in }\Omega \times 0. \end{aligned}$$(12)
Yield criterion and evolution of internal variables
Assumptions on the used free energy
Weak form of the thermomechanical problem
Spatial discretization of the thermomechanical continuum
Remark 1
All algorithms presented later are directly applicable to both finite elements and isogeometric analysis, such that no further distinction will be made in the following. For the sake of brevity, no introduction to isogeometric analysis will be given here, since there has been an overwhelming amount of publications on IGA in the past decade, including the monograph [25]. For details on the application of the dual mortar method to isothermal isogeometric contact mechanics, we refer to our recent work [22].
Time discretization
Remark 2
While being relatively easy to implement and fairly robust, the presented generalized\(\alpha \) time integration schemes are not algorithmically energy conserving. As an alternative, socalled structure preserving time integration schemes based on the (generalized) energy momentum method have been proposed in [33–36] for isothermal nonlinear elasticity. Later, those methods have been extended to isothermal contact [37], elastoplasticity [38], thermoelasticity [39–41] and thermoelastic contact [11]. The combination of the cited works to a structure preserving time integration for a fully coupled thermoelastoplastic contact problem is beyond the scope of this paper, but might be a worthwhile topic of future research.
Solution algorithm using nonlinear complementarity functions
Thermomechanical contact
Having dealt with the thermomechanical coupling within the bulk structure, we turn our focus to thermomechanical contact problems. First, the continuum mechanical description is recalled in “Problem setup” section, which, in more detail, can also be found e.g. in [42, 43]. Next, a weak form extending the small strain formulation presented in [12] is derived in “Weak form of the thermomechanical contact problem” section. Finally, we present the new mortar finite element formulation for finite deformation thermomechanical contact problems in “Mortar finite element discretization of thermomechanical contact” section.
Problem setup
Remark 3
The presented thermal conditions at the contact interface are actually thermodynamically consistent, meaning that they obey the first and second law of thermodynamics. To prove this, the conditions above can be reformulated in terms of a dissipation potential at the interface and a subsequent analysis similar to the one given above for the bulk continuum can be conducted, see e.g. [2, 42] for a detailed derivation. Since this derivation would lead to no further insight with regard to the following numerical algorithms, it is skipped here and the reader is referred to the respective literature instead.
Remark 4
The parameters \(\beta _c\) and \(\delta _c\) also have a direct physical interpretation: The product \(\beta _cp_n\) in (76) and (77) determines the heat flux across the contact interface due to the temperature jump, i.e. it describes the contact heat conductance. In the presented description, which follows [2, 42], the heat flux is a linear function of the contact pressure. However, more involved nonlinear relations can be found in the literature. For instance, [43] distinguishes three sources of heat conduction across rough surfaces: conduction through contacting asperities, heat conduction in enclosed gas and radiation. In the formulation presented later, nonlinear models for heat conduction could be employed simply by replacing the product \(\beta _cp_n\) with a nonlinear relation \(\bar{\beta }_c(p_n)\) or, when formulated using the later defined Lagrange multipliers (representing the negative contact traction), by replacing \(\beta _c\lambda ^c_n\) with \(\bar{\beta }_c(\lambda ^c_n)\) in (95).
The parameter \(\delta _c\) determines how frictional heat is distributed to the two bodies. In the limit cases \(\delta _c=0\) or \(\delta _c=1\) the entire frictional heat is added to the master or slave side, respectively.
Weak form of the thermomechanical contact problem
Mortar finite element discretization of thermomechanical contact
The discrete interpolation of displacements and temperatures at the contact interface follows directly from (43) by restricting the ansatz functions to the element boundary. In the present notation, we will not explicitly distinguish between the ansatz functions in the continuum and at the boundary. Instead, the context will provide the necessary information, i.e. if the quantities are integrated over a volume, the ansatz functions refer to (43) and if integration is performed on the contact surface only the trace space restriction of the ansatz functions are evaluated.
Remark 5
By choosing a fully implicit time discretization of the contact forces instead of some linear combination of \({}^{n+1}{}{{{\mathbf {f}}}}^c_u\) and \({}^{n}{}{{{\mathbf {f}}}}^c_u\), we guarantee that the contact work in one time step \(W^c = ({}^{n+1}{}{{{\mathbf {\mathsf{{d}}}}}}{}^{n}{}{{{\mathbf {\mathsf{{d}}}}}})^\mathrm {T}{}^{n+1}{}{{{\mathbf {f}}}}^c_u\) becomes negative (i.e. dissipative) for nodes that come into contact and zero for nodes leaving the active contact set. Following an idea in [51] the dissipated energy can be reintroduced into the system via a velocity update procedure. If, however, the contact force were discretized by a linear combination of two time steps, nodes leaving the active contact set would introduce energy to the discrete system, which one might not be able to compensate via the velocity update procedure.
Solution algorithm using nonlinear complementarity functions
Numerical results
In the following, several numerical examples are presented to demonstrate the wide range of applications covered with the methods presented above. First, an anisotropic thermoelastoplastic necking problem without contact is analyzed. Next, two examples validate the ability of the thermomechanical contact algorithm to correctly model heat conductance and frictional dissipation at the interface by comparing the results to analytical and numerical reference solutions. After that, energy conservation is investigated in a dynamic thermomechanical contact setting. Whenever material and geometrical parameters do not model a realworld example, but only a numerical testcase, units of measurement will be omitted. Finally, a fully coupled thermoelastoplastic contact simulation concludes this section. All presented algorithms have been implemented in our parallel inhouse multiphysics research code BACI [56].
Thermally triggered necking of an anisotropic circular bar
Thermally triggered necking of an anisotropic circular bar—material parameters
Shear modulus  G  \(80.2\;\mathrm {GPa}\) 
Bulk modulus  \(\kappa \)  \(164.2\;\mathrm {GPa}\) 
Initial yield stress  \(y_0(T)\)  \((1\omega _0(TT_0))\; 450\;\mathrm {MPa} \) 
Anisotropy parameters  \(y_{11}\)  \(y_{11} = \{0.825 y_0,y_0\}\) 
\(y_{22},y_{33},y_{12},y_{13},y_{23}\)  \(y_{22}=y_{33}=y_{12}=y_{13}=y_{23}=y_0\)  
Linear hardening modulus  \(H^i(T)\)  \((1\omega _h(TT_0))\;129.24\;\mathrm {MPa} \) 
Saturation yield stress  \(y_\infty (T)\)  \((1\omega _h(TT_0))\;715\;\mathrm {MPa} \) 
Hardening exponent  \( \delta \)  16.93 
Density  \(\rho _0\)  \(7.8\cdot 10^{9}\;\frac{\mathrm {N}\,\mathrm {s}^2}{\mathrm {mm}^4}\) 
Heat capacity  \(C_v\)  \(3.588\;\frac{\mathrm {N}}{\mathrm {mm}^2\,\mathrm {s}^2\,\mathrm {K}} \) 
Heat conductivity  \(k_0\)  \(45\;\frac{\mathrm {N}}{\mathrm {s}\,\mathrm {K}}\) 
Expansion coefficient  \(\alpha _T\)  \(10^{5}\;\frac{1}{\mathrm {K}}\) 
Yield stress softening  \(\omega _0\)  \(0.002\;\frac{1}{\mathrm {K}}\) 
Hardening softening  \(\omega _h\)  \(0.002\;\frac{1}{\mathrm {K}}\) 
Dissipation factor  \(\beta \)  0.9 
Initial temperature  \(T_0\)  \(293\;\mathrm {K}\) 
surrounding temperature  \(T_\infty \)  \(293\;\mathrm {K}\) 
Convection coefficient  \(h_c\)  \(17.5\;\frac{\mathrm {N}}{\mathrm {mm}\,\mathrm {s}\,\mathrm {K}}\) 
Stationary heat conduction
Frictionless contact with a rigid obstacle: convergence study
Frictional heating of a rotating ring
Bouncing ball
Squeezed elastoplastic tube
Conclusions
In this article, a fully coupled monolithic formulation for thermoelastoplasticity and thermoelastoplastic frictional contact problems has been derived. Following one common theme, the newly developed algorithm deals with all arising inequality constraints of contact, friction and plasticity by using nonlinear complementarity functions. The algorithm for solving the thermoelastoplastic problem in the bulk structure is based on our previous developments in [20], where the plastic constraints are treated via nonlinear complementarity functions at every quadrature point.
In the presented extension to thermoplasticity, the emerging thermomechanical coupling terms within a monolithic solution framework are consistently linearized. By condensation of the plastic variables at quadrature level, the method has the same computational efficiency as the radial return mapping, while, at the same time, being potentially more robust, since the plastic inequality constraints only need to be satisfied at convergence of the global NewtonRaphson method, but not at every step as in the return mapping. Moreover, this general concept can be used to develop other types of complementarity functions, which may result in even more robust and efficient computational schemes.
Concerning thermomechanical frictional contact at finite deformations, a new discretization approach based on dual mortar finite element methods has been derived. The model includes a pressure dependent heat conduction across the contact interface as well as frictional work, which is consistently converted to heat. The use of dual basis functions allows for an easy condensation of the additional Lagrange multiplier degrees of freedom, such that the resulting system is no longer of saddlepointtype as many other contact formula are. Since the final global system to be solved only involves displacement and temperature degrees of freedom, this renders the algorithm very efficient with a quadratic rate of convergence in Newton’s method, yet at the same time very accurate as the obtained spatial convergence results underline. Finally, robust compatibility of the presented methods has been demonstrated in a fully coupled thermoelastoplastic frictional contact setup.
Declarations
Author's contributions
AS developed the idea, conducted numerical experiments and wrote draft. WAW and AP finetuned the research idea, suggested numerical experiments and revised the paper. All authors read and approved the final manuscript.
Acknowledgements
None.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
Funding was provided by the German Research Foundation (Deutsche Forschungsgemeinschaft—DFG) in the framework of the Priority Programme “Reliable Simulation Techniques in Solid Mechanics. Development of Nonstandard Discretisation Methods, Mechanical and Mathematical Analysis” (SPP 1748) under projects PO1883/11 and WA1521/151.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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