Model-Free Data-Driven Inelasticity
We extend the Data-Driven formulation of problems in elasticity of Kirchdoerfer and Ortiz  to inelasticity. This extension differs fundamentally from Data-Driven problems in elasticity in that the material data set evolves in time as a consequence of the history dependence of the material. We investigate three representational paradigms for the evolving material data sets: i) materials with memory, i. e., conditioning the material data set to the past history of deformation; ii) differential materials, i. e., conditioning the material data set to short histories of stress and strain; and iii) history variables, i. e., conditioning the material data set to ad hoc variables encoding partial information about the history of stress and strain. We also consider combinations of the three paradigms thereof and investigate their ability to represent the evolving data sets of different classes of inelastic materials, including viscoelasticity, viscoplasticity and plasticity. We present selected numerical examples that demonstrate the range and scope of Data-Driven inelasticity and the numerical performance of implementations thereof.
Kirchdoerfer and Ortiz [1, 2, 3] and Conti et al.  have recently proposed a new class of problems in static and dynamic elasticity, referred to as Data-Driven problems, defined on the space of strain-stress field pairs, or phase space. The problems consist of minimizing the distance between a given material data set and the subspace of compatible strain fields and stress fields in equilibrium. They find that the classical solutions are recovered in the case of linear elasticity and identify conditions for convergence of Data-Driven solutions corresponding to sequences of material data sets. Data-Driven elasticity effectively reformulates the classical initial-boundary-value problem of elasticity directly from material data, thus bypassing the empirical material modelling step altogether. By eschewing empirical models, material modelling empiricism, modelling error and uncertainty are eliminated entirely and no loss of experimental information is incurred.
It should be noted that the use of material data as a basis for constitutive modeling is classical and remains the subject of extensive ongoing research. There is a vast body of literature devoted to that subject, including recent developments based on statistical learning, model and data reduction, nonlinear regression, and others, which would be too lengthy to enumerate here. It bears emphasis, that what sets the present approach apart from these other approaches is that we reformulate the classical boundary value problems of mechanics, including inelasticity and approximations thereof, directly on the basis of the material data, without any attempt at modeling the data or performing any form of data reduction or manipulation.
A natural extension of the Data-Driven paradigm concerns inelastic materials whose response is irreversible and history dependent. The theory of materials with memory furnishes the most general representation of such materials. According to Rivlin :
”The characteristic property of inelastic solids which distinguishes them from elastic solids is the fact that the stress measured at time t depends not only on the instantaneous value of the deformation but also on the entire history of deformation.”
The origins of the theory may be traced to a series of papers by Green and Rivlin starting in 1957 [6, 7, 8], who proposed the use of hereditary constitutive laws, originally developed by Boltzman  and Volterra  in the linear case, for the description of non-linear viscoelastic materials as an alternative to models using constitutive equations of the rate type . The hereditary functional approach to inelasticity was introduced into thermodynamics by Coleman . A linearization of Green and Rivlin’s theory was developed by Pipkin and Rivlin . Rheological properties of solids often have a fading memory property, enunciated by Truesdell  as:
”Events which occurred in the distant past have less influence in determining the present response than those which occurred in the recent past”.
Other general representations of inelasticity are based on continuum thermodynamics with internal variables (cf., e. g., ). These representations replace an explicit dependence on history by a dependence on the effects of history, i. e., the current microstructure of the material element. The variables used to describe that microstructure are referred to as internal variables. Together with the state of stress or deformation and a thermodynamic variable such as temperature or entropy, they define the local state of a material element. Such models were introduced for viscoelastic deformation by Eckart , Meixner , Biot  and Ziegler , and have been extensively studied since [25, 26, 27, 28, 29, 30]. The foundations underlying the memory-functional and the internal-variable formalisms were critically reviewed by Kestin and Rice . The correspondence and, in some cases, equivalence between the material-with-memory, internal variable and differential formulations of inelasticity have also been extensively investigated [28, 32, 33, 30].
In the context of Data-Driven inelasticity, the representational paradigms just outlined translate into corresponding representational paradigms for the material data set. Specifically, we identify the material data set at time with the collection of stress-strain pairs that are attainable by the material at that time. For inelastic materials, depends on the past history of stress and strain. The central issue of Data-Driven inelasticity thus concerns the formulation of rigorous yet practical representational paradigms for the evolving material data set. The practicality of the representation revolves around the amount of data that needs to be carried, or generated, along with the calculations. By rigorous we specifically mean representations that result, albeit at increasing computational cost, in convergent approximations.
We specifically consider three representational paradigms: i) materials with memory, i. e., conditioning the material data set to the past history of deformation; ii) differential materials, i. e., conditioning the material data set to short histories of stress and strain; and iii) history variables, i. e., conditioning the material data set to ad hoc variables encoding partial information about the history of stress and strain. We also consider combinations of the three paradigms thereof and investigate their ability to represent the evolving data sets of different classes of inelastic materials, including viscoelasticity, viscoplasticity and plasticity. The resulting Data-Driven inelasticity problems then consist of minimizing distance in phase space between the evolving data set and a time-dependent constraint set. We additionally concern ourselves with the numerical implementation and convergence characteristics of the resulting Data-Driven schemes.
We structure the paper as follows. In Section 2 we succinctly summarize the Data-Driven approach to elasticity by way of background and in order to set essential notation. Extensions to inelasticity predicated on various representations of the material data set are put forth and developed in Section 3. In Section 4 we present selected examples of application to viscoelastic solids that demonstrate the suitability of differential representations of the material data set and the performance of the resulting Data-Driven schemes. Further examples of application are presented in Section 5 that demonstrate how hybrid differential/history variable representations of the material data set can be used to account for hardening plasticity. Finally, an extended discussion of possible extensions and alternative approaches is presented in Section 6.
2. Background: Data-Driven elasticity
We begin by recalling the Data-Driven reformulation of elasticity [1, 2] as a basis for subsequent generalizations to inelasticity. For simplicity, we consider discrete, or discretized, systems consisting of nodes and material points. The system undergoes displacements , with and the dimension of the displacement at node , under the action of applied forces , with . The internal state of the system is characterized by local stress and strain pairs , with and the dimension of stress and strain at material point . We regard as a point in a local phase space and as a point in the global phase space .
The internal state of the system is subject to the compatibility and equilibrium constraints of the general form
where are elements of volume and is a discrete strain operator for material point . We note that constraints (1) are universal, or material-independent. They define a subspace, or constraint set,
consisting of all compatible and equilibrated internal states. In (2) and subsequently, the symbol is used to mean ’given’ or ’subject to’ or ’conditioned to’. Within this subspace, the internal state satisfies the power identity
In classical elasticity, the problem (1) is closed by appending local material laws, e. g., functions of the general form
where . However, often material behavior is only known through a material data set of points obtained experimentally or by some other means. Again, the conventional response to this situation is to deduce a material law from the data set by some appropriate means, thus reverting to the classical setting (4).
The Data-Driven reformulation of the classical problems of mechanics consists of formulating boundary-value problems directly in terms of the material data, thus entirely bypassing the material modeling step altogether . A class of Data-Driven problems consists of finding the compatible and equilibrated internal state that minimizes the distance to the global material data set . To this end, we metrize the local phase spaces by means of norms of the form
for some symmetric and positive-definite matrices , with corresponding distance
for . The local norms induce a metrization of the global phase by means of the global norm
with associated distance
for . The distance-minimizing Data-Driven problem is, then,
i. e., we wish to find the point in the material data set that is closest to the constraint set of compatible and equilibrated internal states or, equivalently, we wish to find the compatible and equilibrated internal state that is closest to the material data set .
We emphasize that the local material data sets can be graphs, point sets, ’fat sets’ and ranges, or any other arbitrary set in phase space. Evidently, the classical problem is recovered if the local material data sets are chosen as
We note that, for fixed , the closest point projection onto follows by minimizing the quadratic function subject to the constraints (1). The compatibility constraint (1a) can be enforced directly by introducing a displacement field . The equilibrium constraint (1b) can then be enforced by means of Lagrange multipliers representing virtual displacements of the system. With given, e. g., from a previous iteration, the corresponding Euler-Lagrange equations are 
which define two standard linear displacement problems. The closest point then follows as
A simple Data-Driven solver consists of the fixed point iteration 
for and arbitrary, where denotes the closest point projection in onto . Iteration (13) first finds the closest point to on the material data set and then projects the result back to the constraint set . The iteration is repeated until , i. e., until the data association to points in the material data set remains unchanged.
The convergence properties of the fixed-point solver (13) have been investigated in . The Data-Driven paradigm has been extended to dynamics , finite kinematics  and objective functions other than phase-space distance can be found in . The well-posedness of Data-Driven problems and properties of convergence with respect to the data set have been investigated in .
3. Extension to inelasticity
A natural extension of the Data-Driven paradigm just described concerns inelastic materials whose response is irreversible and history dependent. The equilibrium boundary-value problem for these materials is, therefore, time dependent. For simplicity, we restrict attention to time-discrete formulations and seek to approximate solutions at times , , , , , . In this setting, the compatibility and equilibrium constraints (1) become
where , , and are the displacements, forces, strains and stresses at time , respectively. The constraints (14) define the constraint set
which is now time-dependent on account of the time-dependency of the applied loads.
In addition, the instantaneous response of inelastic materials is characterized by its dependence on the past history of deformation. By virtue of this history dependence, the set of stress-strain pairs attainable at a material point depends itself on time. We specifically define the instantaneous local material data set as
i. e., the set of local stress-strain pairs attainable at time at material point given the past history of the material point. We additionally define a global material data set at time as .
With these definitions, the Data-Driven problem of inelasticity is
i. e., we wish to find the point in the material data set at time that is closest to the constraint set at time or, equivalently, we wish to find the internal state in the constraint set at time that is closest to the material data set at time . Evidently, the inelastic Data-Driven problem (17) represents a natural extension of the elasticity Data-Driven problem (9) in which both the constraint set and the material data set are a function of time.
The central challenge now is to formulate rigorous yet practical means of characterizing the history dependence of the local material data sets , eq. (16). As noted in the introduction, inelastic material behavior can alternatively be described by means of hereditary laws, within the general framework of materials with memory, rheological and thermodynamical models based on internal variables, by means of so-called differential models and by other means. These constitutive formulations give rise to corresponding representational paradigms in the context of Data-Driven inelasticity, which we elucidate next.
3.1. General materials with memory
A general material with memory is a material whose state of stress is a function of the past history of strain, i. e.,
where is the stress at material point and time , is the corresponding history of strain prior to and is a hereditary functional. For linear rheological materials, takes the form of a hereditary or Duhamel integral expressed in terms of a relaxation kernel .
In a discrete setting, (18) can be approximated as
where is the stress at material point at time , is the strain history of material point up to time and is a discrete hereditary function. In this representation, the local material data sets (16) take the form
i. e., they consist of pairs of stress and strain known to be attainable at time given the past history . In particular, we note that the material data set at time depends on the entire history of strain up to and including time .
As noted in the introduction, materials often exhibit a fading memory property whereby their instantaneous behavior is a function primarily of the recent state history and is relatively insensitive to the distant past history. Examples include viscoelastic materials exhibiting relaxation and bounded creep. For those materials, the strain history in (19) can be truncated beyond a certain decay time, which simplifies the parametrization of the local material data sets . These simplifications notwithstanding, keeping track of long deformation histories, and sampling material behavior conditioned to them, may be challenging and onerous even for materials with fading memory.
3.2. Internal variable formalism
Thermodynamic models based on internal variables are often used to characterize inelasticity and history dependence. In these models, the state at a material point is described in terms of, e. g., its strain, temperature and an additional array of auxiliary variables variables, or internal variables. Thermal processes are beyond the scope of this paper and we shall omit explicit reference to temperature and other thermodynamic variables for simplicity.
In order to describe the behavior of the material, we may assume a Helmholtz free energy , with corresponding equilibrium relations
where are thermodynamic driving forces conjugate to and and denote the derivatives of with respect to strain and internal variables, respectively. In addition, the evolution of the internal variables is governed by kinetic relations of the form
where is a dissipation function and its derivative.
In a time-discrete setting, the evolution of the internal variables is governed by incremental kinetic relations, e. g., of the form 
and the stress-strain relations (21a) specialize to
i. e., is the set of all stress and strain pairs accessible to the material given the prior internal state .
3.3. Relation between the internal variable and hereditary representations
which, evidently, is a particular case of (18).
In the time-discrete setting, the internal variable formalism, eqs. (23) and (24), may also be regarded as a means of defining time-discrete hereditary laws of the form (19). Thus, solving (23) for the interval variables gives a relation
where plays the role of a propagator. Inserting into (24), we further obtain the stress-strain relation
conditioned to the prior internal state . Iterating this relation, we obtain
which defines a discrete hereditary law of the form (19) for the stresses as a function of the past history of strain. However, instead of the general history parametrization (20) the material data set now admits the more explicit representation (25), which greatly reduces the complexity of the parametrization of the material data set relative to that based on a general hereditary framework.
3.4. History variables
Despite its appeal, the essential conceptual drawback of the internal variable formalism is that the internal variable set is often not known or is the result of modeling assumptions. The efficiency of the internal variable parametrization can be retained, while eschewing ad hoc modeling assumptions, simply by reinterpreting internal variables as history variables. Contrary to internal variables, history variables need not have a specific physical meaning and their function is simply to record partial information about the history of the material.
By way of motivation, we may iterate the update (28) to obtain the relation
which gives the internal variables at as a function of the strain history up to and including . More generally, we may consider history variables of the form
i. e., functions of the stress and strain histories up to and including . Implicit in the internal variable framework is that the current material data set depends on the deformation history only through a reduced set of history-dependent internal variables , eq. (25).
The paradigm shift now consists of regarding the variables not as physical variables but as ad hoc history variables that record and store partial information about the past internal history of the material point. Thus, the history variables at time are the result of applying ad hoc history functionals to the prior history of stress and strain. The history functionals query that history and extract and record selected information. The history information is then used to condition and parametrize the material data sets as in (25). However, in the new reinterpretation (25) represents the set of all known stress and strain pairs consistent with all past stress and strain histories for which the chosen history functionals evaluate to .
Importantly, the choice of history variables is no longer a matter of material modeling, as is the case for internal variables, but a question of approximation theory. Specifically, the aim is to produce sequences of history functionals that constrain arbitrary histories of stress and strain increasingly tightly, and exactly in the limit. In particular, the sequence of Data-Driven solutions constrained by an increasing number of history variables should converge to the exact Data-Driven solution corresponding to (20). In practice, the central representational challenge is to characterize general material histories to arbitrary accuracy with as few history variables as possible.
3.5. Differential representations
Differential models of inelasticity (cf., e. g., ) offer the advantage of reducing history dependence to short histories of stress and strain. Differential materials are characterized by a differential constraint of the form
between the strain and its first time derivatives and stress and its first derivatives, for some material-specific function taking values in . In a time-discrete setting, the time derivatives are replaced by divided-difference formulas of the form
for some coefficients dependent on the choice of discrete times . For constant time step,
with coefficients independent of as expected. Inserting these formulas into (33), we obtain a relation of the form
between the short histories of strain of length and short histories of stress of length .
In this representation, the local material data sets (16) take the form
i. e., consist of all pairs of stress and strain at time that are attainable, or known to be attainable, to the material element given the past short histories of stress and strain .
We note from (37) that, for differential models, the material data set (37) indeed depends on history through short histories of stress and strain. This parametrization is in contrast with that obtained from general representations of materials with memory, eq. (20), in which the history dependence of the material data set is parameterized in terms of entire, or long, histories of strain only. We thus conclude that conditioning of material data sets by means of both stress and strain histories may result in smaller parameterizations than otherwise required when only strain histories are accounted for. It may also be reasonably expected that increasing the order of differential representations (37) should lead to increasingly accurate, and in the limit exact, representations of broad classes of materials.
3.6. Equivalence between the internal variable and differential formalisms
The correspondence between the internal variable and differential formalisms can be established as follows. For simplicity, we specifically assume internal variables of the form , with . This assumption sets the tensorial character of the internal variables to be that of a collection of internal strains. Begin by writing (21a) as
Assuming sufficient differentiability, we can differentiate this relation with respect to time and combine the result with the kinetic relations (22) to obtain the identity
Iterating this process, we obtain the system of equations
with the functions defined recursively. Assuming solvability, system (40) can be solved for the internal variables to obtain a hereditary relation of the form
Inserting this relation in (38), we obtain the differential constraint
which is of the general form (33).
A similar connection can be forged directly in the time-discrete setting. Thus, iterating the propagator (28), we obtain the system of equations
for . Assuming again solvability, the system (43) can be solved to obtain
which supplies a time-discrete differential representation of the form (36).
We thus conclude that internal variable and differential representations of material behavior are equivalent when the constitutive relations are sufficiently differentiable and the material behavior is stable. As already noted, within a Data-Driven framework the key conceptual advantage of the differential representation is that it relies on fundamental data only, namely, stress and strain data, and the internal variable set, if any, need not be known.
4. Numerical examples: Viscoelasticity
We proceed to illustrate the preceding representational paradigms, and the Data-Driven schemes that they engender, by means of selected examples of application. Viscoelasticity is characterized by the smoothness of the kinetic equations and the existence of a stable equilibrium manifold. The corresponding data sets of viscoelasticity therefore lend themselves ideally to a differential representation, eqs. (33) and (36).
4.1. Example: The Standard Linear Solid
The Standard Linear Solid, consisting of a Maxwell unit in parallel with an elastic unit, provides a simple and convenient example. The Standard Linear Solid Helmholtz free energy is
where is an internal inelastic strain and and are moduli. The corresponding equilibrium relations (21) are
where is the thermodynamic driving force conjugate to . Assuming linear kinetics, we further have
where is a relaxation time.
A straightforward calculation shows that the inelastic strain can be eliminated from the above equations, using the time-derivative of (47a) in addition, and that the resulting differential constraint is
which is of the form (33). A straightforward time discretization further gives
The corresponding differential representation (37) of the data set is
which, for fixed , defines a linear subspace of phase space of dimension . We conclude that first-order differential representations of the data set of the form (37), with , suffice to represent the Standard Linear Solid exactly. More generally, first-order differential representations of the form (37) can only be expected to furnish an approximation of the actual, and unknown, material behavior.
4.2. Example: The relaxation test
We illustrate the Data-Driven problem defined by the Standard Linear Solid by means of the simple example of relaxation test of a bar, Fig. 1. In this case, the solution consists of a single time-dependent stress and strain pair . The constraint set is then constant and simply restricts the strain to be constant and equal to a prescribed value , i. e.,
Inserting this condition into the differential constraint (50), gives the relation
A straightforward calculation gives the Data-Driven solution as
Thus, the initial material data set is a line of slope roughly through the origin that intersects the constraint set at , which is the instantaneous response of the solid. Subsequent material data sets translate downwards in phase space and their intersection with the constraint set traces the relaxation curve of the bar. More general Data-Driven solutions can be obtained if the material data set is allowed to be a point set, e. g., approximating the Standard Linear Solid data set just described. In this case, the Data-Driven solution is the point in the constraint set closest to the material data set . With the passage of time, these points again trace a Data-Driven relaxation curve of the bar, Fig. 1.
4.3. Convergence analysis: Truss structures
We demonstrate the convergence properties of Data-Driven viscoelasticity with the aid of the three-dimensional truss structure shown in Fig. 2. The geometry of the truss, which comprises 1,246 bars, the boundary conditions and the applied loads are also shown in Fig. 2. The loads are linearly ramped up to , subsequently held constant up to , linearly ramped back to zero at , and held again constant up to . The data sets are generated on the fly by randomizing the Standard Linear Solid data set (51). The data points are assumed to be uniformly distributed within a band of width . A typical local material data set is shown in Fig. 1. The resulting material data sets converge uniformly to the Standard Linear Solid graph in the sense defined in . The parameters of the reference Standard Linear Solid used in calculations are , and . In addition, a constant time step is used in all calculations.
Fig. 2(a) depicts displacement histories at the output node shown in Fig. 2 and Fig. 2(b) shows the history of the resultant of the reaction forces at the kinematically constraint nodes, cf. Fig. 2. The convergence of the time histories towards the solution of the reference Standard Linear Solid with increasing number of materials data points is evident in the figures. The rate of convergence can be monitored by means of the weighted error
where is the number of time steps, is as in (7) and is the solution for the reference Standard Linear Solid. Weighted norms such as (55) arise naturally in the analysis of viscoelastic problems (cf., e. g., ). Compiling statistics over independent runs, i. e., with different randomizations of the data set, we arrive at the convergence plot shown in Fig. 4. Remarkably, the computed rate of convergence is quadratic, or twice the linear rate of convergence characteristic of elastic problems .
5. Numerical examples: Plasticity
Plasticity (cf., e. g., ) supplies an example of a class of material data sets that are not amenable to a strict differential representation and require the use of history variables in addition.
5.1. Example: The isotropic-kinematic linear-hardening solid
We illustrate this class of materials by means of the simple isotropic-kinematic linear-hardening solid, Fig. 5. In this case, the free energy is of the form
where is a internal inelastic strain, is an effective accumulated plastic strain, is a stored energy of cold work and and are moduli. The equilibrium relations (21) evaluate to
where is the yield stress. For the rate-independent solid, the dual kinetic potential is of the form
for some convex yield function , i. e., vanishes within the elastic domain and equals elsewhere in driving-force space. We note that is not differentiable and, therefore, the corresponding kinetic relations
which defines a standard convex-optimization problem . Introducing a Lagrange multiplier , the corresponding Euler-Lagrange equations are
subject to the Kuhn-Tucker conditions
which encode the yielding and loading-unloading conditions. A fully-implicit discretization of (61) gives the time-discrete maximum dissipation principle
where are regarded as given. The corresponding Euler-Lagrange equations are
subject to the Kuhn-Tucker loading-unloading conditions
These equations are closed by the time-discrete equilibrium relations
We note that the material data set of points attainable at time is fully characterized by and . The dependence of on is consistent with a differential representation. However, the additional dependence on is typical of a history variable representation. Indeed, the history-variable character of can be revealed as follows. Taking a convenient seminorm of (65a) and eliminating together with (65b), we obtain