Conference Agenda

The treatment of cardiovascular diseases most often involves coronary stents. Even with drug-eluting stents, implantation can give rise to in-stent restenosis: endothelial denudation and overstretch injuries may result in uncontrolled tissue growth and formation of obstruction to the blood flow. Critical areas where such side effects occur highly depend on the shear stresses and drug distribution inside the artery. For this reason, the analysis of blood flow dynamics in stented arteries is of great interest. The current work is aimed at coupling hemodynamics and tissue growth to include the fluid-structure interaction of pharmacokinetics at the interface between artery and lumen.

Navier Stokes equations and Newtonian constitutive model are used to simulate blood in a stented artery. Wall shear stress (WSS) related quantities are analyzed as indicators of the possible areas of inflammation and thrombosis. Drug elution and deposition on the vessel wall is modeled by means of an advection-diffusion equation and tailored boundary conditions [1]. The convective field is obtained coupling the drug equation to a steady averaged blood flow over three heart beats. Since the healing process and drug elution span a time frame of weeks, a staggered approach is derived to simulate the drug release into the blood stream. Advection-diffusion-reaction equations form the basis of modeling the transport and interaction of species in the vessel wall. The corresponding equations for PDGF, TGF-ß, ECM and SMC can be found in [2]. The drug concentration field is coupled at the interface between the arterial wall and the lumen to account for downstream deposition of the drug. All governing equations for the wall species are coupled to a continuum mechanical description of volumetric growth.

In this work, we test our method on a simplified ring stent geometry with matching interface between the artery wall and the blood domain. We compare the effects of drug coupling and WSS on the endothelium and volumetric growth. All simulations are performed by means of finite element method using FEAP and the in-house code XNS.

[1] Hassler S, Ranno AM, Behr M. Finite-element formulation for advection–reaction equations with change of variable and discontinuity capturing. Computer Methods in Applied Mechanics and Engineering, 2020; 369: 113171.

[2] Manjunatha K, Behr M, Vogt F, Reese S. A multiphysics modeling approach for in-stent restenosis: Theoretical aspects and finite element implementation. Computers in Biology and Medicine, 2022; 150: 106166.

In-stent restenosis is one of the main adverse events after initially successful percutaneous coronary interventions (PCI) with stent implantation. Comprehensive statistical analyses of large clinical datasets identified several independent risk factors for restenosis occurrence, such as patient- or lesion-specific factors, which include small vessel size or the extended length of the stented section. However, it is widely accepted that the local mechanical state within the vessel wall strongly affects vascular growth mechanisms. Nevertheless, these biomechanical factors are currently not integrated into the predictive assessment of lesions at risk. For instance, high intramural stresses and overstretch of healthy vascular tissue during PCI may disturb the natural homeostasis and thus promote excessive tissue growth. Additionally, insufficient stent expansion and incomplete stent apposition reduce the long-term success rate of the procedure. We propose an individualized biomechanical model to study the influence of specific plaque characteristics on the mechanical state of the artery wall during loading conditions experienced in PCI and the final stent placement. In this work, we employ patient-specific artery models based on coronary computed tomography angiography data combined with resolved models of the stent delivery system for physics-informed PCI simulations. We define the system as a computational structural mechanics problem with large deformations and a nonlinear, viscoelastic material formulation for the artery considering the plaque constituents in a heterogeneous manner. The stent structure is resolved and is discretized with reduced-dimensional 1D Cosserat continua with an elastoplastic material formulation. An idealized inflatable balloon model governs the stent expansion. The interaction between balloon catheter and artery is modeled with computational contact mechanics using mortar methods; for the stents, we utilize a beam-to-solid contact approach. All simulations are performed with our in-house multiphysics high-performance code BACI, which uses finite element methods for all problem types considered here. We assess the local stresses and strains within the vessel wall during and after the stent implantation and collate cases with different lesion characteristics. We evaluate the contact between stent struts and endothelium for lesions at risk of incomplete stent apposition. Additionally, we compare the results of our resolved approach to a simplified model, where we model the stent as a pure cylinder with similar mechanical characteristics. In the future, insights from such modeling may inform the clinical assessment of lesions considered for stent implantation.

Neuroblastoma (NB) is the most frequent solid cancer of early childhood. It is a type of cancer that is highly representative of the cancer disease itself, since NB is strongly heterogeneous with very diverse clinical courses that may vary from an indolent disease causing little or no harm and exhibiting spontaneous regression, to an aggressive disease with fatal progression. For these reasons, NB is considered a paradigm of cancer disease and an excellent context of application for the validation of novel developments which have the ambition to be of potential application in a large variety of solid cancers.

NB tumours consist of two main cell populations, neuroblasts and Schwann cells, and the current neuroblastoma classification is based on histological criteria, e. g. the quantity of Schwannian stroma. Neuroblasts and Schwann cells are primary interest herein for contribute directly to the mechanical properties of the tissue through the proliferation and death processes. Extracellular matrix also have a principal role in the cell-microenvironmental cross-talk therefore the tumour can promote to a better stage or keep growing.

We here present a phenomenological model which takes into account as detail as possible to better mimic the real tumour behaviour. Our hypothesis proposes that tumour evolution can be attributed to three distinct processes: growth, shrinkage, and remodelling. The biomechanical model is based on the mass and cellular balance equations coupled with elasticity. The multispecies model simulates the effect of the cellular processes that occur during tumour growth and shrinkage, namely proliferation and death.

The biomechanical finite element model of NB tumour growth starts from imaging data derived mainly from MRI sequences. This data comprises the geometry, the initial cellularity distribution and the tumour vasculature evaluation. At the end of the simulation, the results obtained are validated with a second set of imaging data obtained after treatment.

The study simulates three-month chemotherapy using real patient cases, and presents two distinct outcomes: in one of them, the tumour volume was reduced 20% and in the other one, the volume decreased 90%. One of the patients was classified as low-risk, following the International Neuroblastoma Risk Group (INRG) system, whereas the other was classified as intermediate-risk. Differences appeared in the histology analysis, which reveal one tumour with a higher concentration of tumoural cells, and in the radiomic data obtained after image analysis. The model effectively reproduces these varying outcomes following the application of chemotherapy, facilitating the identification of cases in which the treatment may be effective.

Biomechanical models typically contain numerous parameters. Global sensitivity analysis helps identify the most influential and the non-influential parameters, as well as interactions between the parameters.

We show how to apply variance-based global sensitivity analysis to complex biomechanical models. As the method necessitates numerous model evaluations, we utilize Gaussian process metamodels [1] to lessen the computational burden. The approach is illustrated for models of active biomechanical systems by applying it to nanoparticle-mediated drug delivery in a multiphase tumour-growth model [2] and the formation of aneurysms in a model of aortic growth and remodelling [3].

We discover that a small number of full model evaluations suffices to effectively differentiate influential from non-influential parameters, while further evaluations enable the estimation of higher-order interactions. From a biomechanical modeling standpoint, we observe that often a few influential parameters predominantly govern the model output variance. Simultaneously, substantial parameter interactions can exist, emphasizing the necessity for global methods.

Gaussian process-based global sensitivity analysis proves feasible and beneficial for intricate, computationally demanding biomechanical models. Specifically, it can serve as a foundational building block for parameter identification.

[1] Le Gratiet L, Cannamela C, Iooss B. A Bayesian Approach for Global Sensitivity Analysis of (Multifidelity) Computer Codes. J Uncertain Quantif. 2, 336–363. DOI: 10.1137/130926869 (2014).

[2] Wirthl B, Brandstaeter S, Nitzler J, Schrefler BA, Wall WA. Global sensitivity analysis based on Gaussian-process metamodelling for complex biomechanical problems. Int J Numer Meth Biomed Engng. e3675. DOI: 10.1002/cnm.3675 (2023).

[3] Brandstaeter S, Fuchs SL, Biehler J, Aydin RC, Wall WA, Cyron CJ. Global Sensitivity Analysis of a Homogenized Constrained Mixture Model of Arterial Growth and Remodeling. J Elast. 145, 191–221 DOI: 10.1007/s10659-021-09833-9 (2021).

Bones have the ability to adapt their structure and thus their density to external loads. Cancellous bone, which forms the spongy interior of bones, is a microstructural network of rods. Under- or overloading strengthens or narrows these rods, altering the microstructural pattern. In this adaption process, osteocytes act as mechanosensors, activated by mechanical signals and regulating the mechanical adaptation of bone. That is, they communicate with bone-forming or bone-resorbing cells. Thus, bone remodelling at a particular point is triggered by non-local mechanosensors in its vicinity, i.e. the sensors involved act in a specific sphere of influence and not only locally.

In this work, we present a micromorphic approach that extends the established concept of local bone adaption to account for both the non-locality of bone remodelling and the heterogeneous structure of the material without explicitly resolving it within a two-scale approach. Our approach enables a simple implementation in the open source finite element environment deal.II and avoids the need for laborious neighborhood sampling, as is the case with integral approaches, or for higher continuity requirements, as is the case with higher gradient approaches.

Our approach is phenomenological in nature and refers to nominal bone density to be interpreted as a macroscopic measure of the ratio of bone mass to pore volume in the underlying trabecular microstructure. This way, we account for the heterogeneous microstructure of bone by capturing its effect on nominal bone density, but without actually resolving individual trabeculae. Since bone is a living material, in the continuum approach to bone remodelling we apply the theory of open-system thermodynamics, which assumes that there is a mass source corresponding to the change in nominal density over time. The mass source is equated with a mechanical stimulus, comparing the stored energy to an attractor. The attractor can be interpreted as a biological stimulus that drives remodelling. In the local case, the stored energy is a purely local quantity that depends on the macroscopic deformation. In our novel non-local approach, we now extend this by adding a micromorphic and a scale-bridging component to the stored energy. This allows us to account for non-locality with a characteristic length scale, which acts as a measure for the heterogeneous microstructure and a scale-bridging parameter that penalizes the deviation of the micromorphic from a higher gradient model.

The approach is illustrated in depth and its implications are discussed using benchmark examples. In addition, the modeling approach is discussed using long tubular bones and compared with CT images in health and osteoporosis.

Gastric peristalsis refers to the coordinated contraction and relaxation of the muscles in the stomach wall that mixes and grinds food and propels chyme down the digestive tract. Gastric peristalsis is realized by an intricate electromechanical system. We present a computational multiphysics framework for its simulation on patient-specific stomach geometries. It combines a robust gastric electrophysiology model with an active-strain finite elasticity model for the tissue mechanics [1,2,3]. The patient-specific spatially varying parameter distributions are determined by a novel algorithm mapping a two-dimensional parameter distribution function onto a general tube-like surface. The capability of the proposed computational framework for large-scale in silico analyses of gastric electromechanics is demonstrated on patient-specific human stomach models derived from magnetic resonance images. The proposed framework can reproduce essential phenomena on patient-specific stomach geometries, including the entrainment and propagation of stable ICC slow waves as well as the propagation of physiological ring-shaped peristaltic contraction waves. In summary, the presented framework provides a powerful tool to study gastric electromechanics in health and disease. This can enable optimized patient-specific diagnosis and therapy planning.

References

[1] Brandstaeter, S., et al., Computational model of gastric motility with active-strain electromechanics. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2018.

[2] Djabella, K., M. Landau, and M. Sorine, A two-variable model of cardiac action potential with controlled pacemaker activity and ionic current interpretation, in 2007 46th IEEE Conference on Decision and Control. 2007, Institute of Electrical & Electronics Engineers (IEEE). p. 5186 - 5191.

[3] Ruiz-Baier, R., et al., Mathematical modelling of active contraction in isolated cardiomyocytes. Mathematical Medicine and Biology, 2014. 31: p. 259-283.

Human brain tissue shows complex, nonlinear, and time dependent mechanical behavior, and thus presents a significant challenge to those interested in developing accurate constitutive models. Research in our group is focused on better understanding the factors that influence the mechanical response of the tissue. To this end, we combine the mechanical testing of tissue samples from different brain regions under finite deformation in compression, tension and shear with microstructural analyses, continuum mechanics modeling, and finite element simulations. The application of an inverse parameter identification allows us to determine material parameters with a subsequent statistical analysis revealing their regional dependence. Here, we find that the corpus callosum and corona radiata in particular have to be considered as regions with distinct mechanical properties when modeling the whole brain. Furthermore, we analyze the protein content of the tested specimens by enzyme-linked immunosorbent assays and show their correlation with the identified material parameters. These results may motivate and guide the development of microstructurally informed constitutive models that may enable patient-specific predictions.

Traumatic brain injury (TBI) brought on by a severe head impact in a car accident, a fall, or a sports injury results in internal tissue damage beyond recovery. The human brain mainly has two vital tissues; gray matter and white matter. During accidental impact, forces and torques are imparted in the brain tissues to trigger significant local damage. Although the brain can recover from a TBI, the force necessary to cause permanent brain damage is still not fully understood. One aspect of investigating TBI is to provide a mathematical model and a computational framework to identify the level of injury. Mechanical characterization of the brain tissue is essential to understand brain damage caused by TBI. Since 1960, many studies have been done to understand the brain’s mechanical behavior. It is found that brain tissue’s behavior is an incompressible, viscoelastic material and anisotropic material. The human brain’s finite element (FE) models have been utilized to investigate the risk and mechanisms of traumatic brain injuries. Many human brain FE models have been developed. Many different constitutive models have been used for different parts of the human head. Still, there is scope for improvement in constitutive modeling and its finite element implementation. In this work, we present an anisotropic viscoelastic constitutive model and essential equations for finite element simulations. We implemented the constitutive model by ABAQUS UMAT for doing finite element simulations of the human head FE-model for real-life loading cases. Our uniaxial and cyclic loading simulation using UMAT agree with experimental and MATLAB results.

The development of effective and locking free shell elements is intensive topic of research since several decades. Recently, the Hellan-Herrmann-Johnson (HHJ) method for linear Kirchhoff-Love plates has been extended to nonlinear Koiter shells. Therein, the bending moment tensor is introduced as additional unknown to rewrite the fourth order as a second order mixed saddle point problem circumventing the necessity of C1-conforming finite elements. Via hybridization techniques the saddle point translates into a minimization problem again.

The tangential-displacement and normal-normal-stress continuous (TDNNS) method has successfully been applied to linear Reissner-Mindlin plates leading to a shear locking free formulation.

In this talk we present a shear locking free extension of the TDNNS method from linear Reissner-Mindlin plates to nonlinear Naghdi shells by means of a hierarchical approach. Therefore, the HHJ method for Koiter shells is enriched with shearing degrees of freedom, discretized by H(curl)-conforming Nedelec elements. We discuss the small-strain regime leading to the HHJ and TDNNS method for linear Koiter and Naghdi shells. We show how the so-called Regge interpolant can be used in all methods to avoid membrane locking by inserting into the membrane energy term.

Several benchmark examples, implemented in the open-source finite element software NGSolve (www.ngsolve.org), are presented to demonstrate the excellent performance of the proposed shell elements.

We model the formation of wrinkles of an elastic substrate coated with a thin film. The elastic substrate is first stretched, then the film is attached to a part of the substrate boundary in the deformed state. Once the the external force is released, wrinkles form due to the stress mismatch between the two materials. The elastic substrate is modeled using a hyperelastic, homogeneous and isotropic material. The film is modeled using a geometrically exact Cosserat shell. The resulting deformation and microrotation $(varphi, R)$ are a minimizing pair of the combined energy functional

$$

J(varphi, R) = int_{Omega} W_textup{bulk}(nablavarphi) : dV + int_{Gamma_c}W_textup{coss}(nablavarphi_{|_{Gamma_c}}, R) : dS

$$

in the admissible set

begin{align*}

mathcal{A} = Big{&(varphi, R) in W^{1,q}(Omega, mathbb{R}^3) times H^1(Gamma_c,textup{SO(3)}) : Big| :

varphi textnormal{ is a deformation function, }& (varphi, R) textnormal{ fullfill the Dirichlet boundary conditions} Big}

end{align*}

with $q > 3$.

We discretize the problem using Lagrange finite elements for the substrate displacement. For the numerical treatment of the microrotation field, standard Lagrange finite elements cannot be used, as the microrotation field maps to the nonlinear manifold $textnormal{SO}(3)$. We present a generalization of Lagrange finite elements that is suitable for such manifold-valued functions: geometric finite elements.

The resulting finite element spaces are complete and invariant under isometries of the manifold. The best approximation error depends on the mesh size h. We prove the existence of solutions of the discrete coupled model. We compare two Newton-type methods to solve the resulting discrete problem: a Riemannian trust-region method and a Riemannian proximal Newton method.

Numerical experiments show that we can efficiently reproduce wrinkling patterns of coupled systems. Our approach works as well for more complex scenarios like multi-layer systems or systems involving various stress-free configurations.

In many different fields of engineering beam models play a significant role in the efficient simulation of slender structures. The most important model for large deformations is the so-called geometrically exact beam also often referred to as Simo-Reissner beam. The configuration manifold of the beam model is given by special Euclidian group as it describes the position of the centerline as well as the orientation of the beam's cross-section. The partial differential equations describing the behavior of the beam is usually solved with the help of the Finite Element Method (FEM). So it becomes necessary to discretize the special orthogonal group in a finite element sense.

A finite element discretization of the special orthogonal group is rather difficult as the special orthogonal group is not an abelian, additive group but a matrix group under multiplication. Though there exist parametrizations of the orthogonal group, which have an additive structure, they result in path-dependency. This can be overcome by discretizing the group directly by using so-called directors. The directors can be discretized additively, so in a classical finite element sense. This, however, leads to an increase in the number of degrees of freedom

if Lagrange multipliers are used to ensure the orthonormality of the directors. Further, this formulation does not conserve the structure of the manifold at every point of the discretization. A possible remedy could be a projection method via the polar decomposition, which is very costly in numerical terms.

The use of unit quaternions for the parametrization presents an interesting alternative. Even though unit quaternions have a complex mathematical structure, it can easily be ensured that their unit length is conserved after a finite element discretization by normalizing the discretized quaternions. This still allows for a classical additive discretization technique in a finite element sense.

In the literature, it is often shown that the Isogeometric Analysis (IGA) is advantageous over the classical FEM with Lagrangian elements, especially for dynamic problems. We thus apply the IGA to the quaternion formulation of the geometrically exact beam.

We present an efficient, robust, objective, singularity-free, and path independent formulation for director fields based on optimization on manifolds. This approach allows for accurate and efficient computations of director fields that arise in geometrically non-linear structural models such as the Reissner-Mindlin shell model, in material models of Cosserat-type and in micromagnetic simulations. In this contribution, we investigate the influence of interpolation on manifolds on locking as well as the application of element technologies, such as enhanced assumed strains and the discontinuous Galerkin method.

The numerical methods are implemented into the open source code Ikarus (https://ikarus-project.github.io/), which enables rapid algorithm prototyping, even for optimization on manifolds, thus highlighting the user-friendly interface of this software.

The pertinent constraint for director fields requires to retain unit length of the director during deformation, which can be satisfied by interpreting the constraint as a restriction on the design space. By transforming the problem from “constrained optimization on an unconstrained space” to “unconstrained optimization on a constrained space”, the structure of the problem is retained, and the design space is reduced. The transformation to an unconstrained optimization problem on a manifold requires generalization of concepts, such as the incremental update of design variables, to account for living on a manifold instead of living in a linear vector space.

For the interpolation on nonlinear manifolds, we utilize the ideas on geometric finite elements presented by Sander (2012) and Grohs (2011). The combination of element technologies such as enhanced assumed strains with the optimization on manifolds approach promises an efficient and accurate solution method for director fields. Numerical examples are presented in the context of micromagnetics, Reissner-Mindlin shells and three-dimensional beams to demonstrate the efficiency and accuracy of the approach.

Sander, O., Geodesic finite elements on simplicial grids. Int. J. Num. Meth. Engng. (2012) 92:999–1025. https://doi.org/10.1002/nme.4366

Grohs, P., Finite elements of arbitrary order and quasiinterpolation for data in Riemannian manifolds. Seminar for Applied Mathematics, ETH Zürich, (2011). https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-56.pdf

Müller, A., Bischoff, M. A Consistent Finite Element Formulation of the Geometrically Non-linear Reissner-Mindlin Shell Model. Arch Computat Methods Eng (2022). https://doi.org/10.1007/s11831-021-09702-7

Recently, the concept of hierarchic structural element formulations has been developed in the group of the authors with a focus on shear deformable Reissner-Mindlin shell formulations [1], [2]. Via reparametrization of the kinematic variables, these formulations possess distinct degrees of freedom for transverse shear. One effect of this hierarchic parametrization is that the resulting elements are intrinsically free from transverse shear locking.

However, the hierarchic structure can also be exploited for an intrinsically selective mass scaling, i.e., a scaling down of the high shear frequencies, which limit the critical time step although being of minor importance for the structural response, while keeping the low bending dominated branch of the frequency spectrum unaffected. This stands in contrast to conventional mass scaling for shear deformable elements, where total rotational inertia is scaled and, therefore, also bending frequencies are manipulated.

In linear kinematics, the hierarchic parametrization leads to an additive structure throughout the kinematic equations, i.e., a clear separation between a Kirchhoff-Love type bending part and an additional shear part. For nonlinear shell kinematics, the assumption of only small shear rotations was made to preserve this additive structure [3].

In this contribution, we present recent investigations on intrinsically selective mass scaling with hierarchic isogeometric structural element formulations and discuss the effects of transverse shear parametrization in transient problems. Additionally, we critically discuss the necessity of a fully nonlinear treatment of shear deformation parts as described in [4].

References:

[1] R. Echter, B. Oesterle and M. Bischoff, A hierarchic family of isogeometric shell finite elements. Comput. Methods Appl. Mech. Engrg., Vol. 254. pp. 170-180, 2013.

[2] B. Oesterle, E. Ramm and M. Bischoff, A shear deformable, rotation-free isogeometric shell formulation. Comput. Methods Appl. Mech. Engrg., Vol. 307, pp. 235-255, 2016.

[3] B. Oesterle, R. Sachse and E. Ramm and M. Bischoff, Hierarchic isogeometric large rotation shell elements including linearized transverse shear parametrization. Comput. Methods Appl. Mech. Engrg., Vol. 321. pp. 383-405, 2017.

[4] Q. Long, P. B. Bornemann and F. Cirak, Shear-flexible subdivision shells. Int. J. Numer. Meth. Engng., Vol. 90, pp. 1549-1577, 2012.

The critical time step in explicit transient analyses depends on the highest frequency of the discretized system. In case of thin-walled structures discretized by solid or solid-shell elements, the critical time step, which is a key factor for computational efficiency, is limited by the highest frequencies related to thickness stretch of the elements [1].

Selective mass scaling (SMS) concepts aim at scaling down the highest frequencies, while keeping the low frequencies as unaffected as possible. Most established SMS concepts are designed for discretizations composed of solid or solid-shell elements, as can be seen for instance in [1,2]. They are designed such that at least translational inertia is preserved. Accuracy of these SMS concepts can be increased by extending the construction of scaled mass matrices in such a way that, additionally, rotational inertia is preserved. But this increases computational costs in case non-linear analyses including large rotations, since scaled mass matrices are anisotropic and need to be reassembled during simulation. These additional costs do not pay off in most applications.

In this contribution, we present recent investigations on SMS techniques, which are based on a concept from finite element technology, that is the Discrete Strain Gap (DSG) method [3]. We show that these novel SMS concepts naturally preserve both translational and rotational inertia and possess high accuracy. In addition, having non-linear problem classes including large rotations in mind, we show how to develop efficient isotropic DSGSMS concepts which avoid the need for reassembly of scaled mass matrices.

REFERENCES

[1] G. Cocchetti, M. Pagani und U. Perego. Selective mass scaling and critical time-step estimate for explicit dynamics analyses with solid-shell elements. Computers & Structures, Vol. 127, pp. 39–52, 2013.

[2] L. Olovsson, K. Simonsson und M. Unosson. Selective mass scaling for explicit finite element analyses. Int. J. Numer. Meth. Engng., Vol. 63(10), pp. 1436–1445. 2005.

[3] K.-U. Bletzinger, M. Bischoff und E. Ramm, A unified approach for shear-locking-free triangular and rectangular shell finite elements. Computers & Structures, Vol. 75(3), pp. 321-334. 2000.

For the description of the load-bearing behavior of structures, the degree of statical indeterminacy is a fundamental property that formally describes the number of missing equilibrium equations necessary to calculate the internal forces. The formal definition of the degree of statical indeterminacy as one single number neglects the distribution of statical indeterminacy within the structure. This neglect can lead to situations where a structure, which is kinematic in one direction and statically indeterminate in another direction is denoted as statically determinate. Thus, mechanisms, which are relevant in the field of deployable structures, and possibilities for prestressing are overlooked. The redundancy matrix, first described by Bahndorf (1991), quantifies the distribution of statical indeterminacy in the structure.

The redundancy matrix is an idempotent matrix, meaning that its eigenvalues are either zero or one. Associated with the eigenvalue of one, which occurs exactly in the quantity that matches the degree of statical indeterminacy, the respective eigenvectors span a space of incompatible elongations (von Scheven et al. (2021)). This space matches the description of self-stress states, described by Pellegrino and Calladine (1986). The eigenvectors associated with the zero eigenvalues span a space that includes states where prescribed elongations match the total elongations. In this case, displacements are present without imposing normal forces, even in statically indeterminate structures. The information about states of stress-free displacements and displacement-free stresses can e.g. be used in the decision-making process of actuator placement in adaptive civil structures (Wagner et al. (2018)) and it might also be used in the adaption process of deployable structures (Veuve et al. (2017)) or for maintenance issues like monitoring stresses in certain parts of a structure.

This contribution presents examples of using the redundancy matrix in the design and assessment of civil engineering structures, especially of deployable structures.

References:

Bahndorf, J. (1991). Zur Systematisierung der Seilnetzberechnung und zur Optimierung von Seilnetzen. Ph. D. thesis, University of Stuttgart, Stuttgart.

Pellegrino, S. and C. Calladine (1986). Matrix analysis of statically and kinematically indeterminate frameworks. Int. Journal of Solids and Structures 22(4), 409–428.

Veuve, N., A. C. Sychterz, and I. F. Smith (2017, December). Adaptive control of a deployable tensegrity structure. Engineering Structures 152, 14–23.

von Scheven, M., E. Ramm, and M. Bischoff (2021). Quantification of the redundancy distribution in truss and beam structures. Int. Journal of Solids and Structures 213, 41–49.

Wagner, J. L., J. Gade, M. Heidingsfeld, F. Geiger, M. von Scheven, M. Böhm, M. Bischoff, and O. Sawodny (2018). On steady-state disturbance compensability for actuator placement in adaptive structures. at - Automatisierungstechnik 66(8), 591–603.

A better understanding of the stress-deformation behavior of concrete structures under different loading and environmental conditions is inevitable to maintain the structural integrity and to avoid catastrophic failures. In the framework of continuum damage mechanics, several material models have been developed in the past to investigate the constitutive response of concrete under different conditions. It has been observed from the experimental studies that the elastic response and stiffness degradation of concrete are dependent on the orientation of micro-cracks and direction of applied loading. This necessitates the incorporation of damage-induced anisotropy[1] in material models for concrete.

In this regard, an anisotropic damage model that describes the softening response of concrete under different loading conditions is developed applying finite element formulations. A two dimensional damage effect tensor is employed to describe the anisotropic evolution of damage. Damage models that take account of the softening responses are known for their spurious mesh dependencies. An implicit gradient enhancement technique introducing an internal length scale is implemented to overcome the numerical difficulties due to damage localization[2]. The model is verified, calibrated and validated considering various experimental results from the literature including monotonic and cyclic loading cases with different load patterns.

[1] R. Desmorat, F. Gatuingt, F. Ragueneau. Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials. Engineering Fracture Mechanics, 74(10), 1539-1560, (2006).

[2] R.H.J. Peerlings, R. De Borst, W.A.M. Brekelmans and J.H.P. De Vree. Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering, 39, 3391-3403 (1996).

Accumulation of plastic deformation in the surface layer of rails and wheels during many rolling contact loading cycles can result in fatigue crack initiation. The behavior and strength of this highly deformed and anisotropic layer are thus key properties of a rail or wheel material. Establishing crack initiation criteria that account for the properties of the material and are experimentally validated is of great importance in railway engineering.

In this contribution, test results from previously conducted axial-torsion experiments on pearlitic R260 steel specimens have been used to assess the accuracy of available crack initiation criteria as well as to suggest modified criteria. In the experiments, solid test bars were predeformed by torsion under different nominal axial stresses to replicate the anisotropic material in the surface layer of rails. Some of the predeformed specimens were re-machined into a thin-walled tubular shape and then subjected to further cyclic multiaxial loading.

Various crack initiation criteria for rolling-contact situations have been proposed in the literature. However, anisotropy has not been considered in many of them, or they are limited to a specific loading condition, or they are not based on experimental data. In this contribution, we predict the cyclic plasticity and anisotropy evolution during the tests by using a finite strain plasticity model and FEM. Then, by using the obtained stress and strain histories, several crack initiation criteria are evaluated as a post-processing step and further improved by considering the effect of anisotropy.

The recording and evaluation of component life in electric drive systems is considerably complicated by the newly used materials and material compositions. Particularly critical are electrical subcomponents which have beam welded connections made of high-purity copper. Due to the strong coupling between stress and strength as well as the novel material properties, established methods of weld strength analysis cannot be applied without restriction. Therefore, an adapted procedure for the evaluation of the fatigue properties of these junctions is to be developed with the aim of a computational proof of service life under high-frequency vibration loads in the VHCF range.

The aim of the talk is to present the complex and thermally determined properties of the special welding spot and the inherent fatigue properties. On the one hand, the extensive and variable test program in relation to the investigated impact types as well as initial sheet configurations will be discussed. On the other hand, a self-contained methodology is to be presented, which guarantees the transferability of the simulatively determined strength between different welded joints. It is based on the NuMeSis method presented by KAFFENBERGER [1], which evaluates the specific, static notch stress situation based on real measured weld seam geometries of steel components. The transferability of the fatigue strength between different welded joints is then achieved by the combined consideration of the micro-support effect according to NEUBER [2] and the weakest link model according to WEIBULL [3]. The transfer of this method from statically loaded steel welds to high-frequency loaded copper welds requires both the embedding of the method in the methodologies of computational vibration fatigue as well as profound numerical changes of the method. This guarantees an efficient, automated evaluation and the consideration of the special properties of the high-purity copper material. Together with other influencing factors such as the presence of internal defects, this procedure leads to a self-contained evaluation concept for welded copper compounds.

[1] Kaffenberger, Vormwald: Considering size effects in the notch stress concept for fatigue assessment of welded joints, Computational Materials Science 64, 2012, S. 71-78

[2] Neuber: Über die Berücksichtigung der Spannungskonzentration bei Festigkeitsberechnungen, Konstruktion 20 Heft 7, 1968, S. 245-251

[3] Weibull: A statistical theory of the strength of materials, Royal Swedish Institute for Engineering Research, 1939

Computational multiscale methods are well-established tools to predict and analyse material behaviour across scales. They are applied so as to reveal the influence of the underlying microstructure on effective material properties and enable complex multi-physics interactions to be accounted for in simulations. Whereas multiscale approaches for thermo-mechanical problems and electro-active solids have been in the focus of intense research in the past decade, rather few works have so far focused on electrical conductors.

Based on the recent works [1,2] this material class and, in particular, the influence of mechanically-induced microscale defects on the effective conductivity is subject of the present contribution. At the outset of the developments, a quasi-stationary setting is assumed such that Maxwell’s equations reduce to the continuity equation for the electric charge and to Faraday’s law of induction. Scale-bridging relations for the kinematic- and flux-type quantities are established, their consistency with an extended Hill-Mandel condition is shown and a closed-form solution for the effective macroscale conductivity tensor based on the underlying microscale boundary value problem is provided.

In view of the experimental investigations [3,4] the effective conductivity tensor, as a fingerprint of the material microstructure, is of primary interest. To study the applicability of the proposed approach, focused ion beam milling is used in a first step to generate geometrically well-defined microstructures [4]. In a second step, focus is laid on mechanically-induced micro-cracks in metal thin films [3]. Both sets of microstructures are electrically characterised by means of four point probe resistance measurements and analysed by means of the proposed computational multiscale scheme. Good accordance between experiment and simulation is achieved which shows the applicability of the proposed multiscale formulation.

[1] T. Kaiser, A. Menzel, An electro-mechanically coupled computational multiscale formulation for electrical conductors, Archive of Applied Mechanics, 91, 1509–1526 (2021)

[2] T. Kaiser, A. Menzel, A finite deformation electro-mechanically coupled computational multiscale formulation for electrical conductors, Acta Mechanica, 232, 3939–3956 (2021)

[3] T. Kaiser, M.J. Cordill, C. Kirchlechner, A. Menzel, Electrical and mechanical behavior of metal thin films with deformation-induced cracks predicted by computational homogenisation, International Journal of Fracture 231, 233–242 (2021)

[4] T. Kaiser, G. Dehm, C. Kirchlechner, A. Menzel, H. Bishara, Probing porosity in metals by electrical conductivity: Nanoscale experiments and multiscale simulations, European Journal of Mechanics A/Solids, 97, 104777 (2023)

Internal traverse grinding (ITG) with electroplated cBN tools and under high speed conditions if a highly efficient process for the machining of hardened steel components. In ITG, the grinding wheel consists of a conical roughing zone and a cylindrical finishing zone. The tool is fed in axial direction into a revolving workpiece, performing roughing and finishing in a single axial stroke. Due to the process kinematics, the process forces during ITG are dependent on the current material removal rate, which varies during the process. The mechanical compliance of the entire system, consisting of both tool- and workpiece spindle, the workpiece clamping device, and all other components in the flow of force, result in shape deviations of the workpieces after machining.

We recently proposed a multi-scale simulation framework to model ITG with electroplated CBN wheels numerically [1]. A digital grinding wheel, based on real grain geometries obtained from optical measurements, was implemented in a geometric physically-based simulation (GPS) to simulate the engagement of each individual grain during the process. The normal force contributions of each individual grain were modelled by a single-grain force model, which was calibrated against two-dimensional Finite Element Simulations of single grain cuts. By taking into account both the system compliance and the total normal force, the deflection between tool and workpiece was modelled in the GPS.

Based on the simulation results, different compensation strategies for the NC tool path were implemented and compared, and a significant reduction of the shape deviations was achieved.

[1] Tsagkir Dereli T, Schmidt N, Furlan T, Holtermann R, Biermann D, Menzel A. Simulation Based Prediction of Compliance Induced Shape Deviations in Internal Traverse Grinding. Journal of Manufacturing and Materials Processing. 2021; 5(2):60. https://doi.org/10.3390/jmmp5020060

Although isotropic hardening plasticity is the most basic hardening type in material modeling, many existing models rely on an overly simplistic hypothesis: a direct relationship between yield stress and accumulated plasticity. Our recent publication [1] falsifies this hypothesis for one medium carbon steel. Furthermore, we investigated the yield surface evolution and its relation to the accumulated plasticity. Several distortional hardening models in the literature assume a direct relationship between the complete yield surface evolution and accumulated plasticity for uniaxial cyclic loading.

These identified modeling deficiencies underscore the necessity for new constitutive models. To tackle this need, we propose a novel plasticity formulation incorporating neural networks in a thermodynamically consistent framework. We extract and analyze the trained neural networks to identify new constitutive equations with sparse regression techniques, cf. [2]. The complete process results in the discovery of new evolution equations based on the experimental data. A notable finding is new interactions between the hidden state variables in the evolution laws. Based on these equations, we can design new specialized experiments to further understand the interplay between loading type and hardening behavior.

[1] K. A. Meyer and J. Ahlström, “The role of accumulated plasticity on yield surface evolution in pearlitic steel,” Mech. Mater., vol. 179, p. 104582, 2023, doi: 10.1016/j.mechmat.2023.104582.

[2] M. Flaschel, S. Kumar, and L. De Lorenzis, “Unsupervised discovery of interpretable hyperelastic constitutive laws,” Comput. Methods Appl. Mech. Eng., vol. 381, p. 113852, 2021, doi: 10.1016/j.cma.2021.113852.

The field of application of thin glass products is vast including various engineering branches such as e.g. electronics, medical equipment and automobiles. In order to realize a cost-efficient production of surface shapes with high accuracy and complexity, a novel replicative glass processing technique called non-isothermal glass molding has been developed (see [1]). However, the production of thin glass components using this technology still raises the issue of shape distortions, cracks and surface defects of molded parts. Therefore, the experimental investigation and mechanical modeling of glass above the glass transition temperature at finite strains are combined in order to simulate these glass forming processes.

Previous experimental studies have shown that the material behavior can be predicted adequately by the Maxwell model (see e.g. [2]). Based on this viscoelastic formulation (see [3]), the material law used is thermo-mechanically consistent and allows the prediction of rheological effects observed during the experiments. In particular, a stress-dependent relaxation time is used to describe the relaxation behavior and the dissipation generated is also taken into account. Regarding the experimental investigation, isothermal uniaxial compression tests above the glass transition temperature are performed for different strain rates and temperatures. By combining the experimental data with the simulation, a multi-curve-fitting is introduced. This nonlinear optimization lead to suitable material parameters with respect to distinct temperatures.

[1] A.-T. Vu, H. Kreilkamp, O. Dambon, and F. Klocke, Optical Engineering 55(7), 071207 (2016).

[2] T. Zhou, J. Yan, J. Masuda, and T. Kuriyagawa, Journal of Materials Processing Technology 209(9), 4484-4489 (2009)

[3] S. Reese and S. Govindjee, International Journal of Solids and Structures 35(26-27), 3455-3482 (1998)

It is still challenging in the field of topology optimization to optimize structures including the complex real-world material behavior. Nevertheless, the specific material behavior has significant influence on the optimal results. Therefore, we proposed a numerical efficient surrogate model for plasticity including hardening extending the established thermodynamic topology optimization (TTO). Even if the simulation results seem reasonable, experimental validation is still mandatory to ensure feasibility for real-world application.

Thus, we present the validation of the thermodynamic topology optimization including plasticity with hardening by comparison of experiments with optimization results. To this end, topology optimized structures are manufactured by additive manufacturing. The real material behavior needs to be determined from additively manufactured tensile specimens so that the material parameters for the specific hardening are considered during the optimization process. Subsequently, structures with respect to hardening as well as elastically reference are optimized and manufactured. Optimization results with pure elastic and plastic model are compared by experiments to show the importance of including hardening behavior within optimization for real-world application.

Finite element (FE) models are widely used to capture the mechanical behavior of structures. Uncertainties in the underlying physics and unknown parameters of such models can heavily impact their performance. Thus, to satisfy high precision and reliability requirements, the performance of such models is often validated using experimental data. In such model updating processes, uncertainties in the incoming measurements should be accounted for, as well. In this context, Bayesian methods have been recognized as a powerful tool for addressing different types of uncertainties.

Quasi-brittle materials subjected to damage pose a further challenge due to the increased uncertainty and complexity involved in modeling crack propagation effects. In this respect, techniques such as Digital Image Correlation (DIC) can provide full-field displacement measurements that are able to reflect the crack path up to a certain accuracy. In this study, DIC-based full field measurements are incorporated into a finite element model updating approach, to calibrate unknown/uncertain parameters of an ansatz constitutive model. In contrast to the standard FEMU, where measured displacements are compared to the displacements from the FE model response, in the force-version of the standard FEMU, termed FEMU-F, displacements are applied as Dirichlet constraints. This enables the evaluation of the internal forces, which are then compared to measured external forces, thus quantifying the fulfillment of the momentum balance equation as a metric for the model discrepancy. In the present work, the FEMU-F approach is further equipped with a Bayesian technique that accounts for uncertainties in the measured displacements, as well. Via this modification, displacements are treated as unknown variables to be subsequently identified, while they are allowed to deviate from the measured values up to a certain measurement accuracy. To be able to identify many unknown variables; including constitutive parameters and the aforementioned displacements, an approximative variational Bayesian technique is utilized.

A numerical example of a three-point bending case study is presented first to demonstrate the effectiveness of the proposed approach. The parameters of a gradient-enhanced damage material model are identified using noisy synthetic data, and the effect of measurement noise is studied. The ability of the suggested approach on identifying constitutive parameters is then validated using real experimental data from a three-point bending test. The full field displacements required as input to the inference setup are extracted through a digital image correlation (DIC) analysis of the provided raw images.

Metal additive manufacturing technologies, such as wire arc additive manufacturing (WAAM), allow the manufacturing of components with maximum freedom in the geometric design and specifically adjusted functional properties. However, WAAM-produced components possess a very wavy surface that exacerbates the numerical simulation of such components. As a result, common finite element approaches with low order shape functions are not suitable for these simulations.

Instead, the finite cell method is chosen for the simulation of tube-like WAAM-produced specimens under a combined tension-torsion load. First, the contour of the specimens is determined with a portable 3D scanning technique. Then, the mechanical response of the specimens is computed using the finite cell method, where a beforehand calibrated J2-plasticity model is applied. The polynomial order of the integrated polynomials is increased for convergence studies and the mechanical response of the specimens is compared to the experimental results. During the experiments, digital image correlation measurements are performed to compare even the full-field deformation to the simulation results. Here, we choose so-called radial basis functions as a global interpolation technique to obtain the in-plane strains and stretches in the curved surfaces for both experiment as well as simulation. Since the material parameters are determined from experimental tensile testing data, uncertainties in these parameters propagate to the numerical simulations of the tube-like specimens under tension-torsion load. To consider these uncertainties when comparing the experimental with the numerical results, the Gaussian error propagation is applied for estimating the uncertainty in the mechanical response.

Modern 3D bioprinting techniques aim at reproducing a specific tissue composition by extruding a bioink, which is a cluster of stem cells embedded into a hosting gel, into the desired pattern. If the extruded structure is fed suitable nutrients, cell differentiation and growth is initiated. However, prior to activating these processes, the gel must first be converted into a polymer construct to provide support and preferential directions to the successive cellular growth phase. There are many ways to accomplish this melt-to-solid transition, most notably photo-polymerisation. The irradiation of a light with suitable intensity and wavelength triggers chemical processes that induce the cross-linking between polymer chains within the printed material, in a time-evolving scheme of structure formation. Controlling this process holds great importance, since cellular motility and nutrient diffusion are greatly affected by the disposition and orientation of the polymer network. As it currently stands, the 3D printing process briefly described above is well known, but in many instances it is not yet adequately optimised and the influence of a variety of parameters hinders a large-scale production basis. For example, the intensity and direction of the UV light has no standard protocol yet, so the definition of an optimal disposition of the light sources can prove essential in minimising the polymerisation times, hence tissue formation times as a whole. This work intends to ground the choice of selected polymerisation parameters to a rational basis. To achieve this, the relevant Physics of what happens after the melted bio-ink is deposited has been represented through multi-physics Finite Element simulations, where the kinetics of polymer cross-linking has been coupled with finite deformation formulations. Viscoelastic behaviour during polymerisation has also been accounted for. To deal with the highly non-linear differential equations representing the problem, a parametrised custom Finite Element variational formulation has been implemented.

Magneto Active Polymers (MAPs) are composite material that combines micron-sized magnetic particles with an elastomer matrix. These materials are notable for their softness and ability to become stiffer in response to an external magnetic field. MAP is prepared by mixing micron-sized iron particles with an elastomeric matrix (i.e., PDMS), which is one of the varieties of silicon rubber. Here we present a study on the mechanical characterization of magneto active polymers prepared by mixing iron particles with a Polydimethylsiloxane (PDMS) (Ecoflex polymers) matrix. The stiffness of PDMS depends on the mixing ratio of these two components. Tensile and relaxation tests were conducted to characterize the mechanical properties of MAP. The experimental data obtained from these tests were used to calibrate the model for the material and to determine the elastic and viscoelastic constants. The results of the study showed that the MAP exhibited desirable mechanical properties and that the external magnetic field can control its response. The calibrated model effectively predicted the mechanical behavior of the material under different loading conditions. The findings of this study have significant implications for the development of magneto active polymers for various applications, such as in the field of soft robotics, where the material's mechanical properties play a crucial role in the design and operation of soft robots.

Magnetorheological elastomers (MREs) with soft matrices have paved the way for new advancements in the fields of soft robotics and bioengineering. The material response is governed by a complex magneto-mechanical coupling, which necessitates the use of computational tools to guide the design process. However, these computational models typically rely on finite element frameworks that oversimplify and idealize the magnetic source and magnetic boundary conditions (BCs), leading to discrepancies with the actual behavior even at a qualitative level. In this study, we comprehensively examine the impact of magnetic BCs and highlight their significance in the modeling process. We present a magneto-mechanical framework that models the response of soft-magnetic and hard-magnetic MREs under various magnetic fields generated by an idealized magnetic source, a permanent magnet, a coil system, and an electromagnet with two iron poles. Our results demonstrate noteworthy differences in magnetostriction depending on the magnetic source used. Furthermore, we implement a virtual testbed to explore the fracture performance of MREs with remanent magnetic fields. To this end, we prescribe remanent magnetization conditions on rectangular samples, and we add a damage phase-field to model crack propagation. In order to maintain the continuity of the magneto-mechanical fields, the damaged material is designed to exhibit the same behavior as the surrounding air. The results show that remanent magnetization enhances the fracture energy and arrests cracks propagation.

Refs:

[1] Lucarini S, Moreno-Mateos MA, Danas K, Garcia-Gonzalez D. "Insights into the viscohyperelastic response of soft magnetorheological elastomers: Competition of macrostructural versus microstructural players". International Journal of Solids and Structures, Vol. 256, 2022.

[2] Moreno-Mateos MA, Hossain M, Steinmann P, Garcia-Gonzalez D. “Hard magnetics in ultra soft magnetorheological elastomers enhance fracture toughness and delay crack propagation”. Journal of the Mechanics and Physics of Solids, Vol. 173, 2023.

Two decades ago, new experiments accompanied by the modernization of magnetoelastic theory have spawned a great amount of theoretical, numerical but also experimental developments on magnetoelastic composites such as magnetorheological elastomers (MREs). Thanks to extensive research efforts, their coupled magnetoelastic response is well understood nowadays. However, this applies only to MREs and related materials based on sufficiently stiff matrix material. Indeed, as the shear modulus of the matrix material is reduced further and further, magnetoelasticity turns out to be an insufficient theoretical framework at the macroscopic scale as demonstrated in this contribution. Even when neglecting the dissipation in the constituents, one may observe significant dissipation.

In composites based on very soft matrix material that can only store rather small amounts of elastic energy, the magnetic energy may dominate the total energy of the system. Multiple (meta-)stable configurations are the consequence, which render the composite material "magneto-pseudoelastic" even when both the inclusions and the matrix material are practically non-dissipative. While such “magnetodeformal shape-memory” effects can be found in mainly experimental literature, we are not aware of quantitatively predictive simulations in this regard.

In this talk we present ongoing work pushing the limits of finite element and re-meshing technologies in order to render the complicated processes extremely soft MREs accessible by computational means.

Hydrogels are three-dimensional networks of polymer chains that are linked together by chemical and physical crosslinks. They are highly swellable, capable of changing chemical energy to mechanical energy and vice versa. They have unique properties such as low elastic moduli and large deformability. The main constituent of hydrogels are the polymer chains that are highly hydrophilic. When immersed in water they absorb water molecules increasing the volume, resulting in swelling. This generally takes place in three steps: one, diffusion of water into the polymer network, two, relaxation of network chains and three, expansion of the polymer network. Normally hydrogels in the fully swollen state are viscoelastic and rubbery, similar to the biological fluids. These properties make them biocompatible. Thus, hydrogels have found applications in biomedical fields, such as making contact lenses, wound dressings, and tissue engineering. They are also used in fluid control and drug delivery systems. In this work, we focus on free swelling of a hydrogel from dry state to fully swollen state. We take the polyacrylamide (PAAm) hydrogel with degree of swelling Q = 42.5. Further, we use this state as the reference state and apply uniaxial load in tension. We assume that swelling is homogeneous. We focus on the non-linear theory of swelling. We plot the stress versus stretch diagram under uniaxial loading conditions. The model is validated with the available experimental results.

Electroactive polymers are a class of smart materials which change shape when stimulated by an electric field. Typical applications are in the areas of, e.g., robotics, artificial muscles and sensors. For such applications a reliable prediction of properties and performance, including loading and performance limits, is important. The occurrence of damage and fracture has a strong influence on the material behaviour. In this context, this work combines a material model for electroactive polymers with a fracture model.

The behaviour of electroactive polymers is modelled as a quasi-static large strain electro-mechanical material. The model is derived from a potential. The mechanical part of the model is a Neo-Hooke material and the electro-mechanical coupling is described by the relative permittivity. The material parameters are chosen such that the model mimics the behaviour of a soft electroactive polymer. The model is analysed with respect to the physically reasonable response and numerical stability. A phase field model for crack propagation is applied as fracture model. This method describes the crack propagation by means of an additional scalar field, the phase field. This phase field takes values between zero and one, whereby value zero represents undamaged material and value one corresponds to a fully damage state, respectively crack at the particular location. Since the model is used for polymers, the phase field model is adapted to large strains. The electro-mechanically coupled problem is solved within a monolithic scheme. The phase field problem, however, is solved within a staggered algorithm. The crack-propagation turns out to be different for the purely mechanical case as compared to the electro-mechanically coupled case.

The proposed model is implemented in a nonlinear finite element framework. Representative numerical examples are discussed in order to show the applicability of the model.

Soft solids such as silicone gels, with bulk shear modulus ranging from ∼10 to 1000 kPa, often exhibit strongly strain-dependent surface stresses. Moreover, unlike conventional stiffer materials, the effects of surface stress in these materials manifest at length scales of tens of micrometers rather than nanometers. The theoretical framework for modelling such problems envisages a soft hyperelastic bulk on which the infinitesimally thin surface that acts as a `wrapper’, with its own constitutive equation. We will recall the essential features of this theoretical framework and its FE implementation in the first part of this talk.

In the second, we will highlight simple force-twist, torque-twist, and force-extension (force-compression) responses of a soft cylinder held between two inert, rigid plates to demonstrate the role that the parameters in the surface constitutive model play in modulating the overall response of the bulk-surface system.

Finally, we will, through Finite Element simulations, demonstrate the effect of surface elasticity on two problems. The first is a variation of the well-known problem of an axisymmetric liquid capillary bridge between two rigid surfaces, with the liquid replaced by a soft solid. When the associated length scales are small, the shapes of the meniscus of a soft solid capillary with significant surface elasticity exhibits a much richer variety of shapes than a liquid. However, with stretch, the meniscus tends behave like a liquid bridge.

In the second problem, we explore the recent rather counter-intuitive experimental observation that, soft solids, when reinforced with small liquid inclusions, can become stiffer than the matrix material. We perform computational homogenisation on liquid inclusion reinforced soft solids with a view to understand the effect of surface stresses on their overall stiffness and manner in which cracks propagate in them.