Finite Strain Research Articles

Topology optimization of incompressible structures subject to fluid–structure interaction

In this work, an algorithm for topology optimization of incompressible structures is proposed, in both small and finite strain assumptions and in which the loads come from the interaction with a surrounding fluid. The algorithm considers a classical block-iterative scheme, in which the solid and the fluid mechanics problems are solved sequentially to simulate the interaction between them. Several stabilized mixed finite element formulations based on the Variational Multi-Scale approach are considered to be capable of tackling the incompressible limit for the numerical approximation of the solid. The fluid is considered as an incompressible Newtonian fluid flow which is combined with an Arbitrary-Lagrangian Eulerian formulation to account for the moving part of the domain. Several numerical examples are presented and discussed to assess the robustness of the proposed algorithm and its applicability to the topology optimization of incompressible elastic solids subjected to Newtonian incompressible fluid loads.

Finite strain modelling of Shape Memory Alloys in the logarithmic strain space: A comparative study with other finite strain approaches

Finite strain models for Shape Memory Alloys (SMAs) offer significant advantages over small strain models in that they excel in their ability to characterize SMA behaviour under large strain and finite rotations while eliminating the inherent assumptions associated with the small strain framework. The paper at hand presents a numerical model for SMAs at finite strain, employing the logarithmic strain measure to obtain an additive split of the total strain into its elastic and inelastic (transformation) parts. Notably, the model maintains the thermodynamic consistency validated using the Clausius–Duhem inequality. Additionally, we compare this model with two other finite strain SMA models employing logarithmic strain measures: a hypoelastic model, which splits the rate of deformation tensor additively, and a hyperelastic model, which decomposes the deformation gradient multiplicatively. The numerical formulation for the three finite strain models is specifically tailored to address superelasticity in SMAs. It is derived from the established small strain model and seamlessly extended into the domain of finite strain framework. Several numerical examples are carried out to showcase the logarithmic strain space approach applied to SMAs and to compare its response with that of the other aforementioned finite strain models in SMAs. The meticulous examination reveals that the logarithmic strain space approach aligns well with the existing finite strain models, demonstrating its effectiveness and compatibility.

Extraction of uniaxial stress–strain curve from bending test using DIC measurements

This paper presents a novel methodology for the analytical determination of the uniaxial tensile/compressive stress–strain curve derived from the four-point bending test without a pre-assumed constitutive model. While the concept of extracting the stress–strain curve from a bending test is not new, the proposed methodology differs from existing methods by recognising the force equilibrium in a deformed configuration, resulting in a temporal variation of both a bending moment and an axial force. In addition, the deformation of the beam cross-section is taken into account according to the principles of finite strain theory either via Poisson’s ratio or by assuming volume conservation. The crucial data for this methodology are obtained from an experimentally conducted bending test in which the loading force, displacement and strain fields at the top, bottom and front faces of the specimen are measured using a stereo DIC. While the method is completely analytical, it reconstructs the stress–strain curve in discrete form by solving a system of linear equations. While the existing methods for the four-point bending test allow extraction of the stress–strain curve from a single experiment only for strains up to a few per cent, our methodology allows the extraction of the stress–strain curve in tension and compression up to five times the strains obtained with methods that neglect the above features.

Benchmarking the accuracy of higher-order particle methods in geodynamic models of transient flow

Abstract. Numerical models are a powerful tool for investigating the dynamic processes in the interior of the Earth and other planets, but the reliability and predictive power of these discretized models depends on the numerical method as well as an accurate representation of material properties in space and time. In the specific context of geodynamic models, particle methods have been applied extensively because of their suitability for advection-dominated processes and have been used in applications such as tracking the composition of solid rock and melt in the Earth's mantle, fluids in lithospheric- and crustal-scale models, light elements in the liquid core, and deformation properties like accumulated finite strain or mineral grain size, along with many applications outside the Earth sciences. There have been significant benchmarking efforts to measure the accuracy and convergence behavior of particle methods, but these efforts have largely been limited to instantaneous solutions, or time-dependent models without analytical solutions. As a consequence, there is little understanding about the interplay of particle advection errors and errors introduced in the solution of the underlying transient, nonlinear flow equations. To address these limitations, we present two new dynamic benchmarks for transient Stokes flow with analytical solutions that allow us to quantify the accuracy of various advection methods in nonlinear flow. We use these benchmarks to measure the accuracy of our particle algorithm as implemented in the ASPECT geodynamic modeling software against commonly employed field methods and analytical solutions. In particular, we quantify if an algorithm that is higher-order accurate in time will allow for better overall model accuracy and verify that our algorithm reaches its intended optimal convergence rate. We then document that the observed increased accuracy of higher-order algorithms matters for geodynamic applications with an example of modeling small-scale convection underneath an oceanic plate and show that the predicted place and time of onset of small-scale convection depends significantly on the chosen particle advection method. Descriptions and implementations of our benchmarks are openly available and can be used to verify other advection algorithms. The availability of accurate, scalable, and efficient particle methods as part of the widely used open-source code ASPECT will allow geodynamicists to investigate complex time-dependent geodynamic processes such as elastic deformation, anisotropic fabric development, melt generation and migration, and grain damage.

A new model of thermo-mechanical actuation for twisted and coiled polymer muscles with initial curvature

The twisted and coiled polymer (TCP) artificial muscle is one type of novel soft actuator for mimicking natural skeletal muscle that can provide large linear and torsional actuation and energy density. Twisting and coiling are the pivotal steps in fabricating TCP muscles. The influence of twisting on the actuation response of TCP muscles has been extensively investigated recently. However, the influence of coiling remains unclear. Based on the finite strain theory, we establish a new thermo-mechanical actuation model for TCP muscles with initial curvature. The theoretical predictions based on the model align well with the finite element simulations, accurately capturing the actuation response of thermally-activated TCP muscles. It is revealed that twisting contributes positively to the actuation, while coiling has a passive effect. Geometrical parameters, such as the helix radius and helix angle, can effectively regulate the actuation performance of TCP muscles. Furthermore, an optimal bias angle is identified that maximizes both the recovery torque and the linear actuation. This study sheds light on the structural optimization design of TCP muscles.

Mechanical stability of homogeneous holographic solids under finite shear strain

We study the linear stability of holographic homogeneous solids (HHS) at finite temperature and in presence of a background shear strain by means of a large scale quasi-normal mode analysis which extends beyond the hydrodynamic limit. We find that mechanical instability can arise either as a result of a complex speed of sound — gradient instability — or of a negative diffusion constant. Surprisingly, the simplest HHS models are linearly stable for arbitrarily large values of the background strain. For more complex HHS, the onset of the diffusive instability always precedes that of the gradient instability, which becomes the dominant destabilizing process only above a critical value of the background shear strain. Finally, we observe that the critical strains for the two instabilities approach each other at low temperatures. We conclude by presenting a phase diagram for HHS as a function of temperature and background shear strain which shows interesting similarities with the physics of superfluids in presence of background superfluid velocity.

Homogenization of nonlinear Cosserat plate including growth theory when the thickness of the plate and the size of the in-plane heterogeneities are of the same order of magnitude

In this work, we present a new two-scale finite-strain plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. Two scales exist, the macroscopic scale is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. This work aims to propose such a theory for thick plates in a nonlinear setting when the thickness and the size of heterogeneities are of the same order of magnitude. The homogenization theory for large deformation with growth is suitable for the modelization of nearly incompressible plant tissue. This model is suitable for wavy leaves. For thick plates, the transverse normal stress and transverse shearing are modelized at both microscopic and macroscopic levels. At the macroscopic level, we consider a nonlinear Cosserat plate model. At the microscopic level, we impose that the average of contribution of the microscopic displacement to rotation angles is equal to zero. We also deal with the problem of boundary layer problem near the lateral boundary. The model recently proposed by Pruchnicki is valid for thin heterogeneous plates; we present an extension for thick plates that takes into account both transverse normal stress and shearing. This model is equivalent to the first model presented but it involves a second-order derivative of the macroscopic displacement field.

Elastic solids under frictionless rigid contact and configurational force

A homogeneous elastic solid, bounded by a flat surface in its unstressed configuration, undergoes a finite strain when in frictionless contact against a rigid and rectilinear constraint, ending with a rounded or sharp corner, in a two-dimensional formulation. With a strong analogy to fracture mechanics, it is shown that (i.) a path-independent J–integral can be defined for frictionless contact problems, (ii.) which is equal to the energy release rate G associated with an infinitesimal growth in the size of the frictionless constraint, and thus gives the value of the configurational force component along the sliding direction. Furthermore, it is found that (iii.) such a configurational sliding force is the Newtonian force component exerted by the elastic solid on the constraint at the frictionless contact. Assuming the kinematics of an Euler–Bernoulli rod for an elastic body of rectangular shape, the results (i.)–(iii.) lead to a new interpretation from a nonlinear solid mechanics perspective of the configurational forces recently disclosed for one-dimensional structures of variable length. Finally, approximate but closed-form solutions (validated with finite element simulations) are exploited to provide further insight into the effect of configurational forces. In particular, two applications are presented which show that a transverse compression can lead to Eulerian buckling or to longitudinal dynamic motion, both realizing novel examples of soft actuation mechanisms. As an application to biology, our results may provide a mechanical explanation for the observed phenomenon of negative durotaxis, where cells migrate from stiffer to softer environments.

A Poroviscoelastic Model at Finite Strains for a Viscous Compressible Solid Skeleton

In this paper, we propose a constitutive model for describing the poroviscoelastic behavior of fluid-saturated porous solids at finite strains. Rather than a viscous incompressible solid skeleton as assumed in previous works, a viscous compressible solid skeleton is considered in this research. To show the poroviscoelastic effect, we apply the model to the study of the classical Terzaghi’s one-dimensional consolidation problem and compare the numerical results with those from poroelasticity. From the numerical results, the secondary consolidation behavior induced by the creep of the porous solid can be observed. Besides, it is shown that the creep of the solid skeleton causes the delay of pore pressure diffusion.

Advancements in Continuum Mechanics and Electrodynamics by a spacetime geometric approach

Results achieved by the authors in the course of research activity on continuum mechanics and electrodynamics (CME) during the past twenty years are illustrated, revised and discussed. Adoption of a geometric approach leads to renewal of concepts and methods of classical CME and to formulation in Euclid (3+1)D ambient spacetime wherein innovation, clarity and depth of a geometric treatment naturally emerge. The dissemination of novel concepts and methods in CME is not delayable, with critical revisitation of problematic notions, analyses and results still currently on the scene. Material frame indifference, equilibrium in a reference configuration, extremality principles in Dynamics, finite elasticity, chain decomposition of finite strain in elasto-thermo-plasticity, variance of electro-magnetic induction laws under frame changes, action on electric charges moving in a magnetic field, are under the spotlight of innovation and advancement. Fostering basic knowledge of Differential Geometry and application of geometric notions and methods contribute effective tools in formulating meaningful rules, amending misstatements and dimming debates based on vague affirmations.

Sound velocities, elasticity and thermal properties of cBN composite with AlN binder at high pressure

Cubic boron nitride (cBN) has been used as a tool material for several decades due to its high hardness and thermal stability. Various binders are commonly added to cBN to synthesise cBN composites for better cutting performance. However, most studies have only investigated properties such as hardness and bulk modulus, without comprehensively studying elastic properties through experiments. In this study, the sound velocities and elastic properties of cBN with AlN binder were determined by using the ultrasonic interferometry technique. The bulk and shear moduli with their pressure dependence were determined as B0 = 351 (3) GPa, G0 = 344 (3) GPa, B0′ = 4.32 (4) and G0′ = 1.74 (2) using the finite strain equation-of-state approaches. Elasticity-related thermal properties, such as Debye temperature, Grüneisen parameter, thermal conductivity and coefficient of thermal expansion were also estimated. The results show that the elastic properties of the cBN composite with vol. 8.8% AlN-binder are comparable to the pure cBN compact. Additionally, the thermal conductivity and coefficient of thermal expansion have little effect on cBN composite with AlN binder compared to pure cBN.

A robust finite strain isogeometric solid-beam element

In this work, an efficient and robust isogeometric three-dimensional solid-beam finite element is developed for large deformations and finite rotations with merely displacements as degrees of freedom. The finite strain theory and hyperelastic constitutive models are considered and B-Spline and NURBS are employed for the finite element discretization. Similar to finite elements based on Lagrange polynomials, also NURBS-based formulations are affected by the non-physical phenomena of locking, which constrains the field variables and negatively impacts the solution accuracy and deteriorates convergence behavior. To avoid this problem within the context of a solid-beam formulation, the Assumed Natural Strain (ANS) method is applied to alleviate membrane and transversal shear locking and the Enhanced Assumed Strain (EAS) method against Poisson thickness locking. Furthermore, the Mixed Integration Point (MIP) method is used to make the formulation more efficient and robust. The proposed novel isogeometric solid-beam element is tested on several single-patch and multi-patch benchmark problems, and it is validated against classical solid finite elements and first-order Lagrange solid-beam element. The results show that the proposed formulation can alleviate the locking effects and significantly improve the performance of the isogeometric solid-beam element. With the developed element, efficient and accurate predictions of mechanical properties of lattice-based structured materials can be achieved. The proposed solid-beam element inherits both the merits of solid elements e.g. flexible boundary conditions and of the beam elements i.e. higher computational efficiency.

A space-time formulation for time-dependent behaviors at small or finite strains

A space-time formulation for time-dependent behaviors at small or finite strains

In Situ Determination of Thermoelastic Properties of Magnesite at High Pressure and Temperature With Implications to Seismic Detectability of Moderately Carbonated Lithologies in the Earth's Mantle

AbstractSubduction of carbonate‐bearing oceanic plates into Earth's interior recycles carbon from the surface to the deep mantle. The subducted carbonates can significantly affect mantle properties and dynamics. Magnesite is recognized as one of the major potential carbon hosts in the deep mantle because of its stability up to deep lower mantle conditions. However, despite many previous studies on the equation of state and elastic properties of magnesite, large discrepancies still exist for its elastic moduli and their pressure and temperature derivatives. Here we report in situ density and elastic wave velocity measurements on a natural magnesite at simultaneous high pressure‐temperature conditions up to ∼8 GPa–1073 K in a multi‐anvil apparatus using ultrasonic and synchrotron X‐ray techniques. Global fitting of the data set to finite strain equations yields KS0 = 114.0 ± 1.2 GPa, = 4.9 ± 0.3, = (−0.019 ± 0.002) GPa/K, G0 = 68.6 ± 0.4 GPa, G′ = 1.6 ± 0.1, and = (−0.018 ± 0.001) GPa/K. Compared to other major upper mantle and transition zone minerals, magnesite has intermediate VP, the lowest VS, and the lowest density. Thus, magnesite possesses higher VP/VS ratio than other mantle minerals in normal mantle regions, whereas this feature is less pronounced in subduction zone environments. Modeling of the velocity profiles of carbonated lithologies along different geotherms suggests that moderately‐enriched magnesite domains are unlikely to be detected seismically in the Earth's mantle.

Reassessment of Birch's Law on hcp‐Fe From Ultrasonic Sound Velocity Measurement and Implications on the Velocity Profiles of Earth's Inner Core

AbstractWe performed in situ X‐ray diffraction and ultrasonic sound velocity measurements on hcp‐Fe up to 15 GPa, 873 K in a multi‐anvil apparatus. The elastic moduli and their pressure and temperature derivatives were determined by fitting the velocity and density data to the third‐order finite strain equations, yielding KS0 = 169.0(57) GPa, KS0′ = 5.4(6), (∂KS/∂T)P = −0.031(3) GPa/K, G0 = 104.5(27) GPa, G0′ = 1.7(2), (∂G/∂T)P = −0.060(2) GPa/K. Within the experimental P‐T range, we find significant temperature effect on the density‐velocity (ρ − VP or S) relations and caution the use of the temperature‐independent Birch's law for extrapolation to core conditions. Furthermore, temperature‐induced velocity decrease is more significant in VS than in VP, offering a possible explanation for the high Poisson's ratio in the core. Extrapolations based on our results together with previous experimental data suggest that VP of hcp‐Fe aligns with PREM at Earth's core conditions, while Vs and density are approximately 10% and 2.7% higher than PREM, respectively.

Nonlinear topology optimization of flexoelectric soft dielectrics at large deformation

We propose a novel nonlinear topology optimization framework tailored for flexoelectric soft dielectrics undergoing large deformation. A numerical method based on Isogeometric analysis (IGA) is introduced to nonlinear soft dielectrics at finite strain, ensuring the C1-continuity for flexoelectric problems. We outline the process of consistent linearizations and IGA discretizations. Additionally, we introduce an innovative and efficient Strain Density Function (SDF) interpolation scheme for optimizing electromechanical coupling factors (ECFs). In this scheme, the interpolation of electromechanical and hyper-elastic energy terms is grounded in the SIMP model, while a linear material interpolation model is employed for the dielectric energy. Our numerical analysis highlights the remarkable performance of the proposed SDF interpolation scheme in nonlinear electromechanical optimization scenarios. An energy remedy scheme is applied to void regions to eliminate the instability of nonlinear optimization, effectively preventing distortion deformations in low-stiffness elements. Furthermore, we investigate the influence of size effect and larger deformation on the optimization of flexoelectric soft materials. The proposed topology optimization framework adeptly leverages the interplay between size effect and larger deformation, resulting in a notable improvement in ECFs within the optimized structures. Finally, the electromechanical coupling factors (ECFs) of the optimized structures in all the examined cases exhibit enhancements up to 9 times as compared to the reference guess designs.

QuadWire: An extended one dimensional model for efficient mechanical simulations of bead-based additive manufacturing processes

This paper presents the basis of a new mechanical model named QuadWire dedicated to efficient simulations of bead-based additive manufacturing processes in which elongated beads undergoing significant cooling and eigenstrain are assembled to form 3D parts. The key contribution is to use a multi-particular approach containing 4 particles per material point to develop an extended 1D model capable of capturing complex 3D mechanical states, while significantly reducing computation time with respect to conventional approaches. Indeed, 3D models usually require at least 3 to 4 elements across the bead section, which results in fine discretization along the tangential direction to avoid conditioning issues, and therefore very fine mesh of the entire 3D part. In the QuadWire model, the bead height and thickness are internal dimensions, enabling a significantly coarser mesh along the tangential direction. Thus, although the QuadWire has 12 degrees of freedom per material point instead of 3 for classical models, the total number of degrees of freedom is reduced by several orders of magnitude for large parts. The proposed model is classically developed within the framework of the principle of virtual power and standard generalized hyperelastic media (i.e., finite strain theory), which necessitates a thermodynamic analysis. Furthermore, the proposed approach includes native and manageable kinematic constraints between successive beads so that the stress state properly evolves during fabrication. Finite element analysis is used for numerical implementation under infinitesimal strain assumption for the sake of simplicity, and the QuadWire stiffness parameters are optimized so that the mechanical response fit conventional 3D approaches. To validate and demonstrate the capabilities of the proposed strategy, the evolution of displacements and stresses in fused deposition modeling of polylactide is simulated.

A finite strain theory for incompressible rubber-like circular arches with an application

A theory for incompressible rubber-like circular arches, undergoing finite strains and rotations, with an application to the stability of Mooney-Rivlin, doubly fixed, thin beams having initially small circular imperfections under uniformly distributed radial loads, is presented. The essence of considering transverse normal strain is emphasized for an incompressible rubber-like material. The effects of the thickness, initial curvature, and perturbation terms of the pseudo strain components on the buckling loads and central deflections are investigated. The thresholds of buckling are observed to depend on the values of the curvature of the initially imperfect beam for the chosen nonlinear material.

Robust and efficient implementation of finite strain generalized continuum models for material failure: Analytical, numerical, and automatic differentiation with hyper-dual numbers

Generalized continuum models for representing nonlinear material behavior including material failure in the finite strain regime are commonly formulated based on scalar elastic and dissipation potential functions. The evolution of stresses and internal variables, i.e., the material state, is governed by partial derivatives of the potential functions with respect to deformation and stress measures. Furthermore, for application of such models in implicit nonlinear finite element analysis tangent operators, consistent with the numerical integration algorithm, are required. In this work, we study analytical, numerical and automatic differentiation schemes for the implementation of coupled, generalized continuum models considering finite inelastic deformations and fracture. This includes the application of finite difference approximations, complex-step derivative approximations and automatic differentiation based on hyper-dual numbers. For the use of automatic differentiation, a semi-analytical split approach is introduced for increasing the computational efficiency. Based on a comprehensive 2D and 3D finite element study, we demonstrate the superior properties of automatic differentiation compared to numerical differentiation methods with regards to accuracy and robustness. Furthermore, the additional computational cost for automatic differentiation using the proposed split approach becomes negligible for large problems.

The power of modern 3-D visualization of high-resolution terrain models in geologic mapping: Complex fold geometries revealed by 3-D mapping in the Panamint metamorphic complex, eastern California, USA

We use structure from motion–multiview stereo (SM) terrain models developed from ground-based images and images acquired from uncrewed aircraft (aka drones) as a base map for three-dimensional (3-D) mapping on the walls of a deep canyon in the Panamint Range of eastern California, USA. The ability to manipulate the 3-D model with views from arbitrary look directions and broad scale range revealed structures that were invisible to conventional two-dimensional (2-D) mapping because of both the scale of the structures and their exposure on vertical to near-vertical cliff faces. The analysis supports field evidence for four phases of ductile deformation, with only one of the younger phases documented on early geologic maps of the area. The oldest deformational event (D1) produced the main metamorphic fabric and pre-dates Late Cretaceous plutons. This deformation produced a 200–250-m-thick high-strain zone localized along marbles at the top of the Kingston Peak Formation and lower Noonday Formation. Geometric analysis from the model suggests strongly that large sheath folds at scales of 100–300 m are developed within these marbles. Large measured finite strains indicate displacement across this apparent shear zone of at least 4–5 km and displacements of tens of kilometers are allowable, yet the structure is invisible to conventional mapping because the high-strain zone is stratabound. The main fabric shows two clear overprints and a third that is likely an even younger deformation. D2 and D3 generated tight to close, recumbent folds and open to tight, upright folds, respectively, both folding the main foliation with localized development of crenulation cleavages axial planar to the folds. An additional overprint shows no clear cross-cutting relationship with D2 or D3 fabrics and could be a manifestation of either of those events, although the deformation is spatially limited to a narrow shear zone beneath a brittle, dextral-normal fault with the same kinematics as a mylonitic fabric in a Cretaceous granite in the footwall. This observation suggests an extensional, core complex–style deformation to produce this structure. We suggest that 3-D mapping has the potential to revolutionize geologic mapping studies, particularly where steep topography provides 3-D views that are virtually invisible on conventional 2-D maps. Previously bewildering geologic puzzles can be solved by the ability to visualize large cliff exposures from arbitrary angles and map the features in true 3-D at resolutions to the centimeter level. Although this study emphasized intermediate scales imaged by a drone, our methods here are easily extended to larger scales using a crewed aircraft for imaging. We suggest these methods should be used routinely in frontier areas with steep terrain where aviation is already in use for access, but the methods can be employed anywhere steep terrain “hides” major rock exposures on conventional 2-D maps.