Continuum Mechanics Model Research Articles

A simple and general framework for the construction of thermodynamically compatible schemes for computational fluid and solid mechanics

We introduce a simple and general framework for the construction of thermodynamically compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems that satisfy an extra conservation law. As a particular example in this paper, we consider the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that describes the dynamics of nonlinear solids and viscous fluids in one single unified mathematical formalism.A main peculiarity of the new algorithms presented in this manuscript is that the entropy inequality is solved as a primary evolution equation instead of the usual total energy conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total energy conservation is obtained as a mere consequence of the proposed thermodynamically compatible discretization. The approach is based on the general framework introduced in Abgrall (2018) [1]. In order to show the universality of the concept proposed in this paper, we apply our new formalism to the construction of three different numerical methods. First, we construct a thermodynamically compatible finite volume (FV) scheme on collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an edge/face-based correction that makes the numerical flux thermodynamically compatible. Second, we design a first type of high order accurate and thermodynamically compatible discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical fluxes that were already used inside the finite volume schemes. And third, we introduce a second type of thermodynamically compatible DG schemes, in which thermodynamic compatibility is achieved via an element-wise correction, instead of the edge/face-based corrections that were used within the compatible numerical fluxes of the former two methods. All methods proposed in this paper can be proven to be nonlinearly stable in the energy norm and they all satisfy a discrete entropy inequality by construction. We present numerical results obtained with the new thermodynamically compatible schemes in one and two space dimensions for a large set of benchmark problems, including inviscid and viscous fluids as well as solids. An interesting finding made in this paper is that, in numerical experiments, one can observe that for smooth isentropic flows the particular formulation of the new schemes in terms of entropy density, instead of total energy density, as primary state variable leads to approximately twice the convergence rate of high order DG schemes for the entropy density.

Deformation of Gels with Spherical Auxetic Inclusions.

Auxetic metamaterials possess unnatural properties, such as a negative Poisson's ratio, which offers interesting features when combined with traditional materials. This paper describes the deformation behavior of a gel consisting of spherical auxetic inclusions when embedded in a conventional matrix. The auxetic inclusions and conventional matrix were modeled as spherical objects with a controlled pore shape. The auxetic particle had a reentrant honeycomb, and the conventional phase contained honeycomb-shaped pores. The deformation behavior was simulated using various existing models based on continuum mechanics. For the continuum mechanics models-the simplest of which are the Mori-Tanaka theory and self-consistent field mechanics models-the auxetic particle was homogenized as a solid element with Young's modulus and Poisson's ratio and compared with the common composite gel filled with rigid spheres. The finite element analysis simulations using these models were performed for two cases: (1) a detailed model of one particle and its surroundings in which the structure included the design of both the reentrant and conventional honeycombs; and (2) a multiparticle face-centered cubic lattice where both the classic matrix and auxetic particle were homogenized. Our results suggest that auxetic inclusion-filled gels provide an unsurpassed balance of low density and enhanced stiffness.

A New Family of Thermodynamically Compatible Discontinuous Galerkin Methods for Continuum Mechanics and Turbulent Shallow Water Flows

In this work we propose a new family of high order accurate semi-discrete discontinuous Galerkin (DG) finite element schemes for the thermodynamically compatible discretization of overdetermined first order hyperbolic systems. In particular, we consider a first order hyperbolic model of turbulent shallow water flows, as well as the unified first order hyperbolic Godunov–Peshkov–Romenski (GPR) model of continuum mechanics, which is able to describe at the same time viscous fluids and nonlinear elastic solids at large deformations. Both PDE systems treated in this paper belong to the class of hyperbolic and thermodynamically compatible systems, since both satisfy an entropy inequality and the total energy conservation can be obtained as a direct consequence of all other governing equations via suitable linear combination with the corresponding thermodynamic dual variables. In this paper, we mimic this process for the first time also at the semi-discrete level at the aid of high order discontinuous Galerkin finite element schemes. For the GPR model we directly discretize the entropy inequality and obtain total energy conservation as a consequence of a suitable discretization of all other evolution equations. For turbulent shallow water flows we directly discretize the nonconservative evolution equations related to the Reynolds stress tensor and obtain total energy conservation again as a result of the thermodynamically compatible discretization. As a consequence, for continuum mechanics the new DG schemes satisfy a cell entropy inequality directly by construction and thanks to the discrete thermodynamic compatibility they are provably nonlinearly stable in the energy norm for both systems under consideration.

Multi-scale mechanobiological model for skeletal muscle hypertrophy.

Skeletal muscle adaptation is correlated to training exercise by triggering different signaling pathways that target many functions; in particular, the IGF1-AKT pathway controls protein synthesis and degradation. These two functions regulate the adaptation in size and strength of muscles. Computational models for muscle adaptation have focused on: the biochemical description of signaling pathways or the mechanical description of muscle function at organ scale; however, an interrelation between these two models should be considered to understand how an adaptation in muscle size affects the protein synthesis rate. In this research, a dynamical model for the IGF1-AKT signaling pathway is linked to a continuum-mechanical model describing the active and passive mechanical response of a muscle; this model is used to study the impact of the adaptive muscle geometry on the protein synthesis at the fiber scale. This new computational model links the signaling pathway to the mechanical response by introducing a growth tensor, and links the mechanical response to the signaling pathway through the evolution of the protein synthesis rate. The predicted increase in cross sectional area (CSA) due to an 8 weeks training protocol excellently agreed with experimental data. Further, our results show that muscle growth rate decreases, if the correlation between protein synthesis and CSA is negative. The outcome of this study suggests that multi-scale models coupling continuum mechanical properties and molecular functions may improve muscular therapies and training protocols.

Dynamics of a deforming planetary body

Self-gravitating planetary bodies possess non-spherical shapes and various density distributions. The dynamics of such bodies are treated using a rigid-body assumption, where bodies do not deform throughout their motion, or simplified force models and shapes like a sphere. These assumptions make problems simple, though it essentially ignores the response of deformation to translation and rotation, and vice versa, limiting its applicability. Describing the coupling of three dynamics modes (deformation, rotation, and translation) is essential to detailing the dynamical evolution of planetary bodies. However, a continuum mechanics approach such as finite element modeling (FEM) needs additional cautions because ill-defined body forces may cause unrealistic stress and displacement conditions. Here, we propose a theoretical framework for characterizing the dynamics of a deforming body, applicable to the numerical implementation of continuum mechanics models. We introduce the kinetics and mechanics to derive the governing equations representing all the dynamics modes. The advantage of this technique is that the equation of deformation no longer deals with translation and rotation included in body forces, which removes difficulties in computing stress and displacement fields around boundary conditions. We demonstrate this technique by considering a system that consists of two masses connected by a spring and a damper. An example of this application discusses the dynamical response of the Cold Classical Kuiper Belt Object (CCKBO) 486958 Arrokoth to the Sky Crater-forming impact event on its small lobe. When the dissipation quality factor, known as the Q factor, is about 100, a typical value for a rubble pile object, the deformation dissipation timescale is just a few days. Considering the yield stress and strain to be ∼10 kPa and ∼0.01, respectively, we derive a displacement of up to tens of meters and normal stress of up to ∼5 kPa if the impact direction consists of the components along the two lobes. Without it, the rotational state change gives no significant displacement and normal stress; however, the major part of the imparted momentum may be converted to shear.

A first order hyperbolic reformulation of the Navier-Stokes-Korteweg system based on the GPR model and an augmented Lagrangian approach

In this paper we present a novel first order hyperbolic reformulation of the barotropic Navier-Stokes-Korteweg system. The new formulation is based on a combination of the first order hyperbolic Godunov-Peshkov-Romenski (GPR) model of continuum mechanics with an augmented Lagrangian approach that allows to rewrite nonlinear dispersive systems in first order hyperbolic form at the aid of new evolution variables. The governing equations for the new evolution variables introduced by the rewriting of the dispersive part are endowed with curl involutions that need to be taken care of. In this paper, we account for the curl involutions at the aid of a thermodynamically compatible generalized Lagrangian multiplier (GLM) approach, similar to the GLM divergence cleaning introduced by Munz et al. for the divergence constraint of the magnetic field in the Maxwell and MHD equations. A key feature of the new mathematical model presented in this paper is its ability to restore hyperbolicity even for non-convex equations of state, such as the van der Waals equation of state, thanks to the use of an augmented Lagrangian approach and the resulting inclusion of surface tension terms into the hyperbolic flux.The governing PDE system proposed in this paper is solved at the aid of a high order ADER discontinuous Galerkin finite element scheme with a posteriori subcell finite volume limiter in order to deal with shock waves, discontinuities and steep gradients in the numerical solution. We propose an exact solution of our new mathematical model, show numerical convergence rates of up to sixth order of accuracy and show numerical results for several standard benchmark problems, including travelling wave solutions, stationary bubbles and Ostwald ripening in one and two space dimensions.

Insights into the subsurface structure of wrinkle ridges on Mars

Wrinkle ridges are important landforms on Mars and other planetary bodies and result from horizontal tectonic shortening. This study provides insights into the subsurface of selected wrinkle ridge sites, based on exposures provided by steep escarpments and crater slopes that crosscut wrinkle ridges. We mapped the complex fold and fault patterns and measured fault dips, where faults appear planar. The fault pattern underneath wrinkle ridges is diverse and is dominated by reverse faults and thrusts. The predominance of one fault system leads to morphological asymmetries of many wrinkle ridges. The upper tip of the dominant thrust fault is commonly situated beneath the steeper wrinkle ridge slope and may outcrop at its base. Polarity changes of wrinkle ridges along strike go along with a change of the dominant thrust direction. Symmetric wrinkle ridges are formed when a conjugate system of thrusts exists in the subsurface. Several wrinkle ridges display a main thrust fault whose dip angle bends over to a shallower dip at depth. The measurements of average dip for primary thrust and backthrust faults are 37° ± 2° and 28 ± 2°, respectively. In general, most of the examined wrinkle ridges are characterized by a multitude of subsidiary and backthrust faults. With the limitation that the uppermost 500-1000 m of the wrinkle ridges subsurface are accessible in this study and despite the large morphological and structural variability of wrinkle ridges, the kinematics of fault-propagation folds models seem to show a large match to our observations and measurements. Likewise, continuum mechanics models show a good fit to the obtained structural data. Fault-bend fold models and simple buckle folding models seem less compatible with observations.

Continuum mechanics modeling of complex fluid systems following Oldroyd's seminal 1950 work

A review of some key developments in the continuum mechanics – based macroscopic modelling of complex fluid flows, following the pioneering work of Oldroyd in the 1950s will be presented. We start by reviewing the major developments achieved by Oldroyd, namely how his work established rules for consistency in developing continuum stress constitutive models extending the material into objective time derivatives for tensor quantities, with first application the stress tensor in viscoelastic constitutive relationships. We then show how the impact of his work to viscoelastic fluid flows has been amplified from the kinetic theory work of Bird and others who established a firm macromolecular structural connection. The use of objective time derivatives was then extended to include second order conformation tensors and through them the connection was made to internal deformation energy by Marrucci. We then further demonstrate the impact of this work by analysing the models through the microstructural descriptions offered within the non-equilibrium thermodynamics formalism. We then conclude with a brief mentioning of direct applications of the original Oldroyd-B model (through stability analyses, etc.) towards bettering our understanding of the rheological behaviour of complex fluid systems.

MULTI-SCALE NUMERICAL APPROACH TO THE POLYMER FILLING PROCESS IN THE WELD LINE REGION

In this paper, a multi-scale coupling mathematical model is suggested for simulating the polymer filling process in the weld line region on a micro scale. The model considers two aspects: one is the coupling model based on stresses in the whole cavity region; the other is the multi-scale coupling model of continuum mechanics (CM) and the molecular dynamics (MD) in a weldline region. A weak variational formulation is constructed for the finite element method (FEM), which is coupled with the Verlet algorithm based on the domain decomposition technique. Meanwhile, an overlap region is designed so that the FEM and the MD simulations are consistent with each other. The molecular backbone orientation of the whole cavity is illustrated and the position of the weld line is determined by the characteristics of the molecular backbone orientation. Finally, the properties of the polymer chain in the weld line region are studied conformationally and dynamically. The conformational changes and movement process elucidate that the polymer chains undertake stretching, entangling and orientating. Moreover, the effect of the number of chains and melt temperature on the spatial properties of chain conformation are investigated.

Axially symmetric continuum mechanical model for the oscillon

The oscillon is a highly localized dynamical phenomenon occurring in a thin horizontal layer of granular material, which rests on a rigid plate oscillating in the vertical direction. The geometry is axially symmetric and physically resembles a splash of liquid due to a falling drop, except that it continually perpetuates itself and does not generate a spreading wave, as is the case for a liquid splash. The oscillon moves from “peak” to “crater” and “crater” to “peak” such that the time from “peak” to “peak” or “crater” to “crater” is twice the period of the oscillating plate. The physics of granular phenomena is not properly understood, and there is no continuum mechanical theory of granular materials which is widely accepted as accurately describing their behaviour. Here for a free-flowing (cohesion-less) granular material, under axially symmetric conditions, we present a partial continuum mechanical analysis assuming the Coulomb–Mohr yield function and non-dilatant double-shearing theory. We examine small perturbations superimposed upon a purely vertical vibration, and make the assumption that throughout the motion, the lower surface of the layer remains in contact with the rigid metal plate. We show how the temporal dependence, which decouples from the spatial structure, is governed by Mathieu’s equation for the physically relevant case of the rigid plate oscillating sinusoidally, and therefore stability is determined by certain key parameters. We explore a variety of possible forms for spatial dependence. The present axially symmetric analysis complements that presented by the authors for plane strain conditions, and we find, quite remarkably, that apart from constants, both flows are governed by similar fourth-order systems of ordinary differential equations. This means that for both plane strain and axially symmetry, analogous pattern forming conditions can operate.

Configurational forces in ferroelectric structures analyzed by a macromechanical switching model

Polycrystalline ferroelectric ceramics are widely used in sensors, actuators, microelectromechanical systems, etc. If a ferroelectric structure possesses some defects like voids or inhomogeneities, its reliability is reduced, and undesired non-homogeneous local concentrations of the electromechanical fields occur. Under the applied external loading, a domain switching region evolves in the vicinity of defects, which is manifested as a reorientation of the remanent polarization vector. In the current work, the nonlinear electromechanical behavior of ferroelectric ceramics is computed by means of three-dimensional finite element analysis, using the phenomenological continuum mechanics model suggested by Landis (J. Mech. Phys. Solids 50(1):127–152, 2002. https://doi.org/10.1016/s0022-5096(01)00021-7) and numerically implemented by Stark (Int. J. Solids Struct. 80:359–367, 2015. https://doi.org/10.1016/j.ijsolstr.2015.09.004). This constitutive law is combined with user-developed elements in Abaqus commercial code for nonlinear coupled electromechanical analyses. By use of the numerical simulations, the evolution of all field variables, in particular of the polarization, is tracked. In a post-processing step, the configurational forces are computed, which express the thermodynamic driving forces acting on the defect. As a typical defect, we consider a circular void in the ferroelectric structure exposed to an alternating electric field. Additionally to the void, other inhomogeneities, namely, a strip of dissimilar material as well as dielectric and piezoelectric inclusions, are investigated. For all cases, the redistribution and evolution of the configurational forces are studied. Besides the essential findings and methodology achieved in this work, the developed software can serve as a basis for further investigations on the failure of composite smart structures and explicit crack modeling using fracture mechanical concepts.

Stiffness Adaptation of a Hybrid Soft Surgical Robot for Improved Safety in Interventional Surgery.

Minimally invasive instruments are inserted per-cutaneously and are steered toward the desired anatomy. The low stiffness of instruments is an advantage; however, once the target is reached, the instrument usually is required to transmit force to the environment. The main limitation of the constant stiffness is predetermined maneuverability and cap of force transmission. Whereas, a highly flexible device can be safely steered through the body but is not suitable for payload limit, while a highly stiff device can have relatively high loads but cannot be steered in highly tortuous trajectories. To overcome this limitation, an adaptive stiffness soft robot was proposed, and the effects of the chamber pressure on the stiffness of the soft robot were investigated. To this end, a single-chamber pneumatic soft robot with one tendon was designed and fabricated. Afterward, a continuum mechanics model based on the nonlinear Cosserat rod model with hyperelastic material model and large deformation kinematics of the robot was developed. The shooting method solved the model as a boundary value problem with Dirichlet and Neumann boundary conditions. The results of the model showed stiffness adaptation feasibility with simultaneous tendon-driving and pneumatic actuation. Thus, to validate the theoretical findings, a series of experimental studies were performed with pressure in the range of 33 to 44 kPa and tendon tensions in the range of 0 to 2.7 N. The theoretical and experimental results for tip displacement and stiffness showed similar trends with a maximum error of 8.25%.

Simulation and optimization of magnetoelastic thin shells

Magnetoelastic thin shells exhibit great potential in realizing versatile functionalities through a broad range of combination of material stiffness, remnant magnetization intensity, and external magnetic stimuli. In this paper, we propose a novel computational method for forward simulation and inverse design of magnetoelastic thin shells. Our system consists of two key components of forward simulation and backward optimization. On the simulation side, we have developed a new continuum mechanics model based on the Kirchhoff-Love thin-shell model to characterize the behaviors of a megnetolelastic thin shell under external magnetic stimuli. Based on this model, we proposed an implicit numerical simulator facilitated by the magnetic energy Hessian to treat the elastic and magnetic stresses within a unified framework, which is versatile to incorporation with other thin shell models. On the optimization side, we have devised a new differentiable simulation framework equipped with an efficient adjoint formula to accommodate various PDE-constraint, inverse design problems of magnetoelastic thin-shell structures, in both static and dynamic settings. It also encompasses applications of magnetoelastic soft robots, functional Origami, artworks, and meta-material designs. We demonstrate the efficacy of our framework by designing and simulating a broad array of magnetoelastic thin-shell objects that manifest complicated interactions between magnetic fields, materials, and control policies.

Effects of stripe domains on the loading-unloading hysteretic behavior of monodomain nematic elastomers

The stress-director coupling of monodomain nematic elastomers may lead to soft deformation and the appearance of stripe domains under uniaxial tension. Based on a continuum mechanical model and finite element method, we investigated the loading-unloading hysteretic behavior of thin monodomain nematic elastomer sheets. Under orthogonal uniaxial loading tests, stripe domains appeared in the center of the specimens after the critical strain was reached. Furthermore, a stress peak was observed, which was followed by a stress plateau. During unloading, these stripe domains did not disappear, and the unloading curve was significantly lower than the loading curve, resulting in large stress-strain hysteresis and unloading residual strain. Additionally, the presence of stripe domains led to an increased strain rate correlation in stress-strain curves, while significantly extending the period of strain recovery without unloading constraints.

On the Fidelity of Computational Models for the Flow of Milled Loblolly Pine: A Benchmark Study on Continuum-Mechanics Models and Discrete-Particle Models

The upstream of bioenergy industry has suffered from unreliable operations of granular biomass feedstocks in handling equipment. Computational modeling, including continuum-mechanics models and discrete-particle models, offers insightful understandings and predictive capabilities on the flow of milled biomass and can assist equipment design and optimization. This paper presents a benchmark study on the fidelity of the continuum and discrete modeling approaches for predicting granular biomass flow. We first introduce the constitutive law of the continuum-mechanics model and the contact law of the coarse-grained discrete-particle model, with model parameters calibrated against laboratory characterization tests of the milled loblolly pine. Three classical granular material flow systems (i.e., a lab-scale rotating drum, a pilot-scale hopper, and a full-scale inclined plane) are then simulated using the two models with the same initial and boundary conditions as the physical experiments. The close agreement of the numerical predictions with the experimental measurements on the hopper mass flow rate, the hopper critical outlet width, the material stopping thickness on the inclined plane, and the dynamic angle of repose, clearly indicates that the two methods can capture the critical flow behavior of granular biomass. The qualitative comparison shows that the continuum-mechanics model outperforms in parameterization of materials and wall friction, and large-scale systems, while the discrete-particle model is more preferred for discontinuous flow systems at smaller scales. Industry stakeholders can use these findings as guidance for choosing appropriate numerical tools to model biomass material flow in part of the optimization of material handling equipment in biorefineries.

A Mechanics Analysis of Carbon Fiber Plain-Woven Thermoset Prepreg during Forming Process Considering Temperature Effect.

The preforming quality of carbon fiber plain-woven thermoset prepreg (CFPWTP) is critical to the performance of composite aerospace parts. The deformation ability of the CFPWTP material during preforming is affected by both the fabric woven structure and the resin viscosity, which is different from the dry textile material. Incorrect temperature parameters can enlarge the resin’s viscosity, and high viscosity can inhibit fiber deformation and cause defects. This study proposes an equivalent continuum mechanics model considering its temperature–force behavior. Picture frame tests and axial tensile tests at 15 °C, 30 °C, and 45 °C are conducted to obtain the temperature–stress–strain constitutional equations. By Taylor’s expansion formula and surface fitting method, the constitutive modulus of the material is obtained. Consequently, a saddle-shaped forming simulation is carried out, which is later validated by experiments. Results show that the accuracy of the predicted model is high, with 0.9% of width error and 5.1% of length error separately. Besides, the predicted wrinkles are consistent with the test in fold position and in deformation trend under different temperatures.

Analytical solutions of bending analysis and vibration of rectangular nano laminates with surface effects

In this paper, a novel continuum mechanics model is developed to investigate the influence of surface effects on the bending and free vibration behaviors of Kirchhoff rectangular nano laminates. The governing equations are established based on the Gurtin–Murdoch theory and the analysis of the equilibrium of forces. Analytical solutions for bending deflections and natural frequencies are presented for selected boundary conditions. Numerical results show that the influences of surface effects on bending deflections and natural frequencies are related to the aspect ratio of the nano laminates and the value of residual surface stress. Moreover, the influence of surface effects can be ignored when the thickness of the nano laminate is large enough.

The simulation of two-dimensional plane problems using ordinary state-based peridynamics

Abstract The ordinary state-based peridynamics (OSB PD) model is an integral nonlocal continuum mechanics model. And the three-dimensional OSB PD model can deal with linear elastic solid problems well. But for plane problems, the calculation results of existing models have large deviations. In this paper, a set of OSB PD models for plane problems is established by theoretical derivation. First, through the strain energy density function equivalence of peridynamics and classical continuum mechanics, the equivalent coefficients of the plane strain and plane stress problems of OSB PD are deduced. Then, consider the cantilever beam deformation simulation under concentrated load. The simulation results show that the maximum displacements are in good agreement with the corresponding analytical solutions in all directions. Finally, in the simulation of the slab with a hole, the two cases of uniform displacement and uniform load are considered, respectively. The simulation results are consistent with the ANSYS analysis results, and the deviation is small, which verifies the validity of the model.

Dynamic compaction of aluminum with nanopores of varied shape: MD simulations and machine-learning-based approximation of deformation behavior

We compare two machine-learning-based approaches, artificial neural network (ANN) and micromechanical model with automatic Bayesian identification of the model parameters, in application to mimicking the deformation behavior of nanoporous aluminum extracted from molecular dynamics (MD) simulations. Reference data are generated by means of MD simulation of both hydrostatic and uniaxial deformation with compression of representative volume elements of aluminum single crystal with nanopores of spherical, cubic and cylindrical shapes at the temperatures of 300, 500, 700 and 900 K. Several typical sizes of the nanopores are considered: The smallest one corresponds to the initial porosity less than 1%, while the largest one gives the initial porosity in the range of 30–50%. Plastic collapse of nanopores in all cases occurred by the mechanism of emission of partial Shockley dislocation half-loops from the pore surface. The emission of initial loop occurred earlier in the case of cylindrical pore; however, this did not lead to an explosive increase in the number of dislocations in the system at this stage. In general, flat free surfaces of pores are less subjected to the dislocation nucleation than the rounded ones. A new physically-based micromechanical model of the plastic compaction of nanoporous metal is formulated with accounting of different pore shapes and anisotropy of the compaction process. Both tested machine-learning approaches show an adequate approximation of MD data. The developed ANN and parameterized micromechanical model are applied to simulate the propagation of a shock wave in nanoporous aluminum in comparison with direct MD simulations of this process; this comparison shows an adequate description of the shock wave structure by means of both approaches incorporated into continuum mechanics modeling. Thus, the developed machine-learning-based approaches can be applied as constitutive equations of nanoporous aluminum in macroscopic simulations of the dynamic compaction and shock-wave processes in this material.

Accelerating the distance-minimizing method for data-driven elasticity with adaptive hyperparameters

Data-driven constitutive modeling in continuum mechanics assumes that abundant material data are available and can effectively replace the constitutive law. To this end, Kirchdoerfer and Ortiz proposed an approach, which is often referred to as the distance-minimizing method. This method contains hyperparameters whose role remains poorly understood to date. Herein, we demonstrate that choosing these hyperparameters equal to the tangent of the constitutive manifold underlying the available material data can substantially reduce the computational cost and improve the accuracy of the distance-minimizing method. As the tangent of the constitutive manifold is typically not known in a data-driven setting, and as it can also change during an iterative solution process, we propose an adaptive strategy that continuously updates the hyperparameters on the basis of an approximate tangent of the hidden constitutive manifold. By several numerical examples we demonstrate that this strategy can substantially reduce the computational cost and at the same time also improve the accuracy of the distance-minimizing method.