Christian Linder is an Associate Professor of Civil and Environmental Engineering and, by courtesy, of Mechanical Engineering. In his research he advances modeling aspects, numerical algorithms, and visualization tools to improve the performance and reliability of simulations to (i) understand physical mechanisms in materials, (ii) create innovative sustainable building materials and structures, and (iii) enable upscaled devices and engineered systems of the environment. In-house (iv) computational method development in the area of Computational Mechanics and Computational Materials Science constitutes the foundation of our research.

Dr. Linder received his Ph.D. in Civil and Environmental Engineering from UC Berkeley, an MA in Mathematics from UC Berkeley, an M.Sc. in Computational Mechanics from the University of Stuttgart, and a Dipl.-Ing. degree in Civil Engineering from TU Graz. Before joining Stanford in 2013 he was a Junior-Professor of Micromechanics of Materials at the Applied Mechanics Institute of Stuttgart University where he also obtained his Habilitation in Mechanics. Notable honors include a Fulbright scholarship, the 2013 Richard-von-Mises Prize, the 2016 ICCM International Computational Method Young Investigator Award, the 2016 NSF CAREER Award, and the 2019 Presidential Early Career Award for Scientists and Engineers (PECASE).

Academic Appointments

Honors & Awards

  • 2019 Presidential Early Career Award for Scientists and Engineers, White House Office of Science and Technology (2019)
  • 2016 NSF CAREER Award, National Science Foundation (2016)
  • 2016 ICCM Young Investigator Award, International Conference on Computational Methods (2016)
  • Richard-von-Mises Prize, International Association of Applied Mathematics and Mechanics (2013)
  • Haythornthwaite Research Initiation Award, ASME Applied Mechanics Division. (2013)

Professional Education

  • Habilitation, University of Stuttgart, Mechanics (2012)
  • PhD, UC Berkeley, Computational Mechanics (2007)
  • MA, UC Berkeley, Mathematics (2006)
  • MSc, University of Stuttgart, Computational Mechanics of Materials and Structures (2003)
  • Dipl.-Ing., Graz University of Technology, Civil and Environmental Engineering (2001)

2020-21 Courses

Stanford Advisees

  • Doctoral Dissertation Reader (AC)
    Qi Zhang, Yang Zhao, Andy Ziccarelli
  • Postdoctoral Faculty Sponsor
    Michael Kraus
  • Doctoral Dissertation Advisor (AC)
    Prajwal Kammardi Arunachala, Yitao Qiu
  • Master's Program Advisor
    Sina Abrari Vajari, Francisco Alberdi, Claire Killian, Liangtong Lyu, Peyton Rice, Zhe Yan, Susanna van de Graaf
  • Doctoral (Program)
    Tim Ngo

All Publications

  • A non-affine micro-macro approach to strain-crystallizing rubber-like materials JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS Rastak, R., Linder, C. 2018; 111: 67–99
  • A variational framework to model diffusion induced large plastic deformation and phase field fracture during initial two-phase lithiation of silicon electrodes COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING Zhang, X., Krischok, A., Linder, C. 2016; 312: 51-77
  • A highly stretchable autonomous self-healing elastomer NATURE CHEMISTRY Li, C., Wang, C., Keplinger, C., Zuo, J., Jin, L., Sun, Y., Zheng, P., Cao, Y., Lissel, F., Linder, C., You, X., Bao, Z. 2016; 8 (6): 619-625

    View details for DOI 10.1038/NCHEM.2492

    View details for Web of Science ID 000376529000020

  • On the enhancement of low-order mixed finite element methods for the large deformation analysis of diffusion in solids INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Krischok, A., Linder, C. 2016; 106 (4): 278-297

    View details for DOI 10.1002/nme.5120

    View details for Web of Science ID 000373348900002

  • Tri-layer wrinkling as a mechanism for anchoring center initiation in the developing cerebellum SOFT MATTER Lejeune, E., Javili, A., Weickenmeier, J., Kuhl, E., Linder, C. 2016; 12 (25): 5613-5620


    During cerebellar development, anchoring centers form at the base of each fissure and remain fixed in place while the rest of the cerebellum grows outward. Cerebellar foliation has been extensively studied; yet, the mechanisms that control anchoring center initiation and position remain insufficiently understood. Here we show that a tri-layer model can predict surface wrinkling as a potential mechanism to explain anchoring center initiation and position. Motivated by the cerebellar microstructure, we model the developing cerebellum as a tri-layer system with an external molecular layer and an internal granular layer of similar stiffness and a significantly softer intermediate Purkinje cell layer. Including a weak intermediate layer proves key to predicting surface morphogenesis, even at low stiffness contrasts between the top and bottom layers. The proposed tri-layer model provides insight into the hierarchical formation of anchoring centers and establishes an essential missing link between gene expression and evolution of shape.

    View details for DOI 10.1039/c6sm00526h

    View details for Web of Science ID 000378935000013

    View details for PubMedID 27252048

  • Computational aspects of growth-induced instabilities through eigenvalue analysis COMPUTATIONAL MECHANICS Javili, A., Dortdivanlioglu, B., Kuhl, E., Linder, C. 2015; 56 (3): 405-420
  • All-electron Kohn-Sham density functional theory on hierarchic finite element spaces JOURNAL OF COMPUTATIONAL PHYSICS Schauer, V., Linder, C. 2013; 250: 644-664
  • Effect of electric displacement saturation on the hysteretic behavior of ferroelectric ceramics and the initiation and propagation of cracks in piezoelectric ceramics JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS Linder, C., Miehe, C. 2012; 60 (5): 882-903
  • The maximal advance path constraint for the homogenization of materials with random network microstructure PHILOSOPHICAL MAGAZINE Tkachuk, M., Linder, C. 2012; 92 (22): 2779-2808
  • A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS Linder, C., Tkachuk, M., Miehe, C. 2011; 59 (10): 2134-2156
  • Finite elements with embedded strong discontinuities for the modeling of failure in solids INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Linder, C., Armero, F. 2007; 72 (12): 1391-1433

    View details for DOI 10.1002/nme.2042

    View details for Web of Science ID 000251949400002

  • Modeling mechanical inhomogeneities in small populations of proliferating monolayers and spheroids. Biomechanics and modeling in mechanobiology Lejeune, E., Linder, C. 2018; 17 (3): 727–43


    Understanding the mechanical behavior of multicellular monolayers and spheroids is fundamental to tissue culture, organism development, and the early stages of tumor growth. Proliferating cells in monolayers and spheroids experience mechanical forces as they grow and divide and local inhomogeneities in the mechanical microenvironment can cause individual cells within the multicellular system to grow and divide at different rates. This differential growth, combined with cell division and reorganization, leads to residual stress. Multiple different modeling approaches have been taken to understand and predict the residual stresses that arise in growing multicellular systems, particularly tumor spheroids. Here, we show that by using a mechanically robust agent-based model constructed with the peridynamic framework, we gain a better understanding of residual stresses in multicellular systems as they grow from a single cell. In particular, we focus on small populations of cells (1-100 s) where population behavior is highly stochastic and prior investigation has been limited. We compare the average strain energy density of cells in monolayers and spheroids using different growth and division rules and find that, on average, cells in spheroids have a higher strain energy density than cells in monolayers. We also find that cells in the interior of a growing spheroid are, on average, in compression. Finally, we demonstrate the importance of accounting for stochastic fluctuations in the mechanical environment, particularly when the cellular response to mechanical cues is nonlinear. The results presented here serve as a starting point for both further investigation with agent-based models, and for the incorporation of major findings from agent-based models into continuum scale models when explicit representation of individual cells is not computationally feasible.

    View details for PubMedID 29197990

  • Area of lineal-path function for describing the pore microstructures of cement paste and their relations to the mechanical properties simulated from mu-CT microstructures CEMENT & CONCRETE COMPOSITES Han, T., Zhang, X., Kim, J., Chung, S., Lim, J., Linder, C. 2018; 89: 1–17
  • Microstructural origin of resistance-strain hysteresis in carbon nanotube thin film conductors PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA Jin, L., Chortos, A., Lian, F., Pop, E., Linder, C., Bao, Z., Cai, W. 2018; 115 (9): 1986–91


    A basic need in stretchable electronics for wearable and biomedical technologies is conductors that maintain adequate conductivity under large deformation. This challenge can be met by a network of one-dimensional (1D) conductors, such as carbon nanotubes (CNTs) or silver nanowires, as a thin film on top of a stretchable substrate. The electrical resistance of CNT thin films exhibits a hysteretic dependence on strain under cyclic loading, although the microstructural origin of this strain dependence remains unclear. Through numerical simulations, analytic models, and experiments, we show that the hysteretic resistance evolution is governed by a microstructural parameter [Formula: see text] (the ratio of the mean projected CNT length over the film length) by showing that [Formula: see text] is hysteretic with strain and that the resistance is proportional to [Formula: see text] The findings are generally applicable to any stretchable thin film conductors consisting of 1D conductors with much lower resistance than the contact resistance in the high-density regime.

    View details for PubMedID 29440431

  • Computational aspects of morphological instabilities using isogeometric analysis COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING Dortdivanlioglu, B., Javili, A., Linder, C. 2017; 316: 261-279
  • A highly stretchable, transparent, and conductive polymer. Science advances Wang, Y., Zhu, C., Pfattner, R., Yan, H., Jin, L., Chen, S., Molina-Lopez, F., Lissel, F., Liu, J., Rabiah, N. I., Chen, Z., Chung, J. W., Linder, C., Toney, M. F., Murmann, B., Bao, Z. 2017; 3 (3)


    Previous breakthroughs in stretchable electronics stem from strain engineering and nanocomposite approaches. Routes toward intrinsically stretchable molecular materials remain scarce but, if successful, will enable simpler fabrication processes, such as direct printing and coating, mechanically robust devices, and more intimate contact with objects. We report a highly stretchable conducting polymer, realized with a range of enhancers that serve a dual function: (i) they change morphology and (ii) they act as conductivity-enhancing dopants in poly(3,4-ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS). The polymer films exhibit conductivities comparable to the best reported values for PEDOT:PSS, with over 3100 S/cm under 0% strain and over 4100 S/cm under 100% strain-among the highest for reported stretchable conductors. It is highly durable under cyclic loading, with the conductivity maintained at 3600 S/cm even after 1000 cycles to 100% strain. The conductivity remained above 100 S/cm under 600% strain, with a fracture strain of 800%, which is superior to even the best silver nanowire- or carbon nanotube-based stretchable conductor films. The combination of excellent electrical and mechanical properties allowed it to serve as interconnects for field-effect transistor arrays with a device density that is five times higher than typical lithographically patterned wavy interconnects.

    View details for DOI 10.1126/sciadv.1602076

    View details for PubMedID 28345040

  • Modeling tumor growth with peridynamics. Biomechanics and modeling in mechanobiology Lejeune, E., Linder, C. 2017


    Computational models of tumors have the potential to connect observations made on the cellular and the tissue scales. With cellular scale models, each cell can be treated as a discrete entity, while tissue scale models typically represent tumors as a continuum. Though the discrete approach often enables a more mechanistic and biologically driven description of cellular behavior, it is often computationally intractable on the tissue scale. Here, we adapt peridynamics, a theoretical and computational approach designed to unify the mechanics of discrete and continuous media, for the growth of biological materials. The result is a computational model for tumor growth that can represent either individual cells or the tissue as a whole. We take advantage of the flexibility provided by the peridynamic framework to implement a cell division mechanism, motivated by the fact that cell division is the mechanism driving tumor growth. This paper provides a general framework for implementing a new tumor growth modeling technique.

    View details for DOI 10.1007/s10237-017-0876-8

    View details for PubMedID 28124191

  • Quantifying the relationship between cell division angle and morphogenesis through computational modeling. Journal of theoretical biology Lejeune, E., Linder, C. 2017; 418: 1-7


    When biological cells divide, they divide on a given angle. It has been shown experimentally that the orientation of cell division angle for a single cell can be described by a probability density function. However, the way in which the probability density function underlying cell division orientation influences population or tissue scale morphogenesis is unknown. Here we show that a computational approach, with thousands of stochastic simulations modeling growth and division of a population of cells, can be used to investigate this unknown. In this paper we examine two potential forms of the probability density function: a wrapped normal distribution and a binomial distribution. Our results demonstrate that for the wrapped normal distribution the standard deviation of the division angle, potentially interpreted as biological noise, controls the degree of tissue scale anisotropy. For the binomial distribution, we demonstrate a mechanism by which direction and degree of tissue scale anisotropy can be tuned via the probability of each division angle. We anticipate that the method presented in this paper and the results of these simulations will be a starting point for further investigation of this topic.

    View details for DOI 10.1016/j.jtbi.2017.01.026

    View details for PubMedID 28119022

  • Highly stretchable polymer semiconductor films through the nanoconfinement effect SCIENCE Xu, J., Wang, S., Wang, G. N., Zhu, C., Luo, S., Jin, L., Gu, X., Chen, S., Feig, V. R., To, J. W., Rondeau-Gagne, S., Park, J., Schroeder, B. C., Lu, C., Oh, J. Y., Wang, Y., Kim, Y., Yan, H., Sinclair, R., Zhou, D., Xue, G., Murmann, B., Linder, C., Cai, W., Tok, J. B., Chung, J. W., Bao, Z. 2017; 355 (6320): 59-?


    Soft and conformable wearable electronics require stretchable semiconductors, but existing ones typically sacrifice charge transport mobility to achieve stretchability. We explore a concept based on the nanoconfinement of polymers to substantially improve the stretchability of polymer semiconductors, without affecting charge transport mobility. The increased polymer chain dynamics under nanoconfinement significantly reduces the modulus of the conjugated polymer and largely delays the onset of crack formation under strain. As a result, our fabricated semiconducting film can be stretched up to 100% strain without affecting mobility, retaining values comparable to that of amorphous silicon. The fully stretchable transistors exhibit high biaxial stretchability with minimal change in on current even when poked with a sharp object. We demonstrate a skinlike finger-wearable driver for a light-emitting diode.

    View details for DOI 10.1126/science.aah4496

    View details for PubMedID 28059762

  • A highly stretchable autonomous self-healing elastomer. Nature chemistry Li, C., Wang, C., Keplinger, C., Zuo, J., Jin, L., Sun, Y., Zheng, P., Cao, Y., Lissel, F., Linder, C., You, X., Bao, Z. 2016; 8 (6): 618-624


    It is a challenge to synthesize materials that possess the properties of biological muscles-strong, elastic and capable of self-healing. Herein we report a network of poly(dimethylsiloxane) polymer chains crosslinked by coordination complexes that combines high stretchability, high dielectric strength, autonomous self-healing and mechanical actuation. The healing process can take place at a temperature as low as -20 °C and is not significantly affected by surface ageing and moisture. The crosslinking complexes used consist of 2,6-pyridinedicarboxamide ligands that coordinate to Fe(III) centres through three different interactions: a strong pyridyl-iron one, and two weaker carboxamido-iron ones through both the nitrogen and oxygen atoms of the carboxamide groups. As a result, the iron-ligand bonds can readily break and re-form while the iron centres still remain attached to the ligands through the stronger interaction with the pyridyl ring, which enables reversible unfolding and refolding of the chains. We hypothesize that this behaviour supports the high stretchability and self-healing capability of the material.

    View details for DOI 10.1038/nchem.2492

    View details for PubMedID 27219708

  • Understanding geometric instabilities in thin films via a multi-layer model. Soft matter Lejeune, E., Javili, A., Linder, C. 2016; 12 (3): 806-816


    When a thin stiff film adhered to a compliant substrate is subject to compressive stresses, the film will experience a geometric instability and buckle out of plane. For high film/substrate stiffness ratios with relatively low levels of strain, the primary mode of instability will either be wrinkling or buckling delamination depending on the material and geometric properties of the system. Previous works approach these systems by treating the film and substrate as homogenous layers, either consistently perfectly attached, or perfectly unattached at interfacial flaws. However, this approach neglects systems where the film and substrate are uniformly weakly attached or where interfacial layers due to surface modifications in either the film or substrate are present. Here we demonstrate a method for accounting for these additional thin surface layers via an analytical solution verified by numerical results. The main outcome of this work is an improved understanding of how these layers influence global behavior. We demonstrate the utility of our model with applications ranging from buckling based metrology in ultrathin films, to an improved understanding of the formation of a novel surface in carbon nanotube bio-interface films. Moving forward, this model can be used to interpret experimental results, particularly for systems which deviate from traditional behavior, and aid in the evaluation and design of future film/substrate systems.

    View details for DOI 10.1039/c5sm02082d

    View details for PubMedID 26536391

  • A micromechanical model with strong discontinuities for failure in nonwovens at finite deformations INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES Raina, A., Linder, C. 2015; 75-76: 247-259
  • The reduced basis method in all-electron calculations with finite elements ADVANCES IN COMPUTATIONAL MATHEMATICS Schauer, V., Linder, C. 2015; 41 (5): 1035-1047
  • A Complex Variable Solution Based Analysis of Electric Displacement Saturation for a Cracked Piezoelectric Material JOURNAL OF APPLIED MECHANICS-TRANSACTIONS OF THE ASME Linder, C. 2014; 81 (9)

    View details for DOI 10.1115/1.4027834

    View details for Web of Science ID 000355556000006

  • Three-dimensional finite elements with embedded strong discontinuities to model failure in electromechanical coupled materials COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING Linder, C., Zhang, X. 2014; 273: 143-160
  • A homogenization approach for nonwoven materials based on fiber undulations and reorientation JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS Raina, A., Linder, C. 2014; 65: 12-34
  • A homogenization approach for nonwoven materials based on fiber undulations and reorientation Journal of the Mechanics and Physics of Solids. Accepted for publication Raina, A., Linder, C. 2014
  • A marching cubes based failure surface propagation concept for three-dimensional finite elements with non-planar embedded strong discontinuities of higher-order kinematics INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Linder, C., Zhang, X. 2013; 96 (6): 339-372

    View details for DOI 10.1002/nme.4546

    View details for Web of Science ID 000325687400001

  • A strong discontinuity approach on multiple levels to model solids at failure COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING Linder, C., Raina, A. 2013; 253: 558-583
  • Modeling reorientation phenomena in nonwoven materials with random fiber network microstructure. Raina, A., Linder, C. 2013
  • 3D finite elements to model electromechanical coupled solids at failure. Linder, C. 2013
  • An analysis of the exponential electric displacement saturation model in fracturing piezoelectric ceramics. Technische Mechanik. Linder, C. 2012; 32: 53-69
  • Homogenization of random elastic networks with non-affine kinematics. Tkachuk, M., Linder, C.  2012
  • New three-dimensional finite elements with embedded strong discontinuities to model solids at failure. Zhang, X., Linder, C.  2012
  • Modeling quasi-static crack growth with the embedded finite element method on multiple levels. Raina, A., Linder, C. 2012
  • All-electron calculations with finite elements. Schauer, V., Linder, C. 2012
  • New finite elements with embedded strong discontinuities for the modeling of failure in electromechanical coupled solids COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING Linder, C., Rosato, D., Miehe, C. 2011; 200 (1-4): 141-161
  • A strong discontinuity based adaptive refinement approach for the modeling of crack branching. Raina, A., Linder, C. 2011
  • Microstructural driven computational modeling of polymers. Tkachuk, M., Linder, C. 2011
  • Finite element solution of the Kohn-Sham equations. Schauer, V., Linder, C. 2011
  • Modeling crack micro-branching using finite elements with embedded strong discontinuities. Raina, A., Linder, C. 2010
  • Numerical simulation of dynamic fracture using finite elements with embedded discontinuities INTERNATIONAL JOURNAL OF FRACTURE Armero, F., Linder, C. 2009; 160 (2): 119-141
  • Finite elements with embedded branching 20th Annual Robert J Melosh Conference Linder, C., Armero, F. ELSEVIER SCIENCE BV. 2009: 280–93
  • Numerical modeling of dynamic fracture. Armero, F., Linder, C. 2009
  • New finite elements with embedded strong discontinuities in the finite deformation range COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING Armero, F., Linder, C. 2008; 197 (33-40): 3138-3170
  • Numerical simulation of dynamic fracture using finite elements with embedded discontinuities. Report No. UCB/SEMM-2008/01, Department of Civil and Environmental Engineering Armero, F., Linder, C.  2008
  • On configurational compatibility and multiscale energy momentum tensors JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS Li, S., Linder, C., Foulk, J. W. 2007; 55 (5): 980-1000
  • Recent developments in the formulation of finite elements with embedded strong discontinuities IUTAM Symposium on Discretization Methods for Evolving Discontinuities Armero, F., Linder, C. SPRINGER. 2007: 105–122
  • New finite elements with embedded strong discontinuities for the modeling of failure in solids. Ph.D. Thesis, Department of Civil and Environmental Engineering Linder, C. 2007
  • Application of differential topology for the derivation of compatibility conservation laws in mechanics. M.A. Thesis, Department of Mathematics, University of California Linder, C. 2006
  • Finite elements with strong discontinuities. Qualifying Report, Department of Civil and Environmental Engineering Linder, C. 2005
  • Analogy model for the axisymmetric elastic edge bending problem in shells of revolution based on Geckeler’s approximation. Guggenberger, W., Linder, C.  2004
  • An arbitrary Lagrangian-Eulerian finite element formulation for dynamics and finite strain plasticity models. M.Sc. Thesis, Computational Mechanics of Materials and Structures, University of Stuttgart. Linder, C. 2003
  • Elastic stress analysis of axisymmetric discontinuities in shells of revolution by an effective ring analogy model. Guggenberger, W., Linder, C. 2003