Finite Strain Research Articles

Effects of lithium insertion induced swelling of a structural battery negative electrode

Structural battery composites fall under the category multifunctional materials with the ability to simultaneously store electrical energy and carry mechanical load. While functioning as the negative electrode, the carbon fibres also act as mechanical reinforcement. Lithium ion insertion in the carbon fibres is accompanied by a large radial expansion of 6.6 % and an axial expansion of 0.85 % of the fibres. Furthermore, the elastic moduli of the carbon fibres are significantly affected by the insertion of lithium. Current structural battery modelling approaches do not consider these features. In this paper, we investigate the effect of lithium insertion in carbon fibres on the structural electrode mechanical properties by developing a computational model considering finite strains and lithium concentration dependent fibre moduli. The computational model enables representation of morphological change, whereby, features such as internal stress state, homogenized tangent stiffness and effective expansion of the electrode caused by carbon fibre lithiation can be predicted. The adopted finite strain formulation allows for consistent consideration of measurement data at varying state of lithiation. The significance of adopting the finite strain formulation is also shown numerically. Finally, by implementing a novel approach to homogenized stress-free expansion, it is shown that the computed expansion of the structural electrode follows a similar trend to what is observed from experiments.

Finite strain HFGMC analysis of damage evolution in nonlinear periodic composite materials

Finite strain HFGMC analysis of damage evolution in nonlinear periodic composite materials

Modelling of damage and failure within polymeric adhesives

AbstractWithin this contribution, we propose a fully thermo‐mechanical coupled isotropic damage model for polymeric adhesives under finite strains. This model is based on the multiplicative decomposition of the deformation gradient into mechanical and thermal parts. We consider rate‐dependent damage behaviour (e.g. creep damage) of the material by applying a Perzyna‐type ansatz for the damage evolution equations. To overcome the pronounced mesh dependencies which would result from the usage of a local damage model, we make use of a gradient‐extended damage formulation. Besides the main aspects of model, we show the thermodynamically consistent derivation of constitutive quantities as well as the numerical treatment of the governing equations. Finally, we show selected numerical examples to demonstrate the capabilities of the model.

Nonlinear Free Vibration of Spinning Pre-Twisted Functionally Graded Material Plates in Thermal Environment

This paper investigates the nonlinear free vibration of spinning pre-twisted functionally graded material (FGM) plates in thermal environment. Based on the Kirchhoff plate theory and von Kármán large-deformation theory, the nonlinear dynamic model is established via the Lagrange equation, where the warping effect is taken into account. The material properties are related to temperature and change according to a power-law exponent distribution. The theoretical solution of nonlinear-free vibration is obtained using the multiple scale method. The effects of power-law exponent, temperature difference, spinning speed, hub radius ratio, and twist angle on nonlinear-free vibration characteristics of the spinning pre-twisted cantilever FGM plates are discussed. The results show that the spinning pre-twisted cantilever FGM plate exhibits nonlinear hardening-spring vibration behavior, while the characteristics weaken with the increase of spinning speed. The twist angle and setting angle have coupling effects on the nonlinear vibration characteristics of spinning pre-twisted FGM plates.

Topology optimization of compliant mechanisms under transient thermal conditions

This work considers multi-material topology optimization of compliant mechanisms under transient thermal and quasi-static mechanical conditions wherein thermally actuated devices are optimized for different operating conditions. The materials are modeled using finite strain thermo-hyperelasticity and a two way coupling between the energy balance and equilibrium equations is investigated. The design updates are generated from the gradient-based method of moving asymptotes optimizer and the sensitivities are computed using the time dependent adjoint sensitivity analysis. Results show the impact of designing for short versus long actuation times.

Bubbles as high-frequency rheometers for soft matter characterization

Precise knowledge of bubble dynamics in tissue is essential for optimizing both ultrasound imaging and ultrasound therapy. Great efforts have therefore been made, over the past decade, to model the behavior of (micro)bubbles in viscoelastic materials. This has led to a series of models based on modified Rayleigh–Plesset equations. Last to date, we propose a novel generalized model build on finite strain theory and relaxation functions. At this stage however, the bottle neck remains our limited knowledge of tissue viscoelasticity at high strain rates, which is a necessary input to any theoretical model. Indeed, traditional rheometers cannot reach frequencies above 1kHz, which is 3 orders of magnitude below the desired MHz frequencies. We will show how ultrasound-driven bubbles can be used as microscale rheometers. More specifically, we use the volumetric oscillations of (monodisperse) phospholipid-coated microbubbles embedded in hydrogels to measure the storage and loss moduli in the MHz frequency range. In addition, we can bridge the gap between kHz and MHz using the surface modes of cylindrical microbubbles trapped in micropits. These oscillations can be both modelled and simulated, and provide access to the effective surface tension and viscosity in the range of 50 to 500 kHz.

Bespoke two-dimensional elasticity and the nonlinear analogue of Cauchy’s relations

Is it possible to design an architectured material or structure whose elastic energy is arbitrarily close to a specified continuous function? This is known to be possible in one dimension, up to an additive constant (Dixon et al., Bespoke extensional elasticity through helical lattice systems, Proc. R. Soc. A. (2019)). Here, we explore the situation in two dimensions. Given (1) a continuous energy function E ( C ) , defined for two-dimensional right Cauchy–Green deformation tensors C contained in some compact set and (2) a tolerance ϵ > 0 , can we construct a spring-node unit cell (of a lattice) whose energy is approximately E , up to an additive constant, with L ∞ -error no more than ϵ ? We show that the answer is yes for affine E s (i.e., for energies E that are quadratic in the deformation gradient), but that the general situation is more subtle and is related to the generalisation of Cauchy’s relations to nonlinear elasticity.

A multi-scale model combining martensitic transformations with multi-phase crystallographic slip

Mechanically induced martensitic transformations are an essential mechanism influencing the constitutive behaviour of many advanced steel grades, such as austenitic stainless steels and Transformation Induced Plasticity (TRIP) alloys. In this work, a single-crystal constitutive model is developed to predict their complex microstructural behaviour during mechanical loading. The model is based on the work of Vieira de Carvalho et al. [16] and crucially proposes crystallographic slip in the martensitic phase after transformation from retained austenite. Austenite and martensite plasticity are incorporated following a standard finite strain single crystal plasticity approach and are fully coupled with the martensite formation kinematics, enabling the simultaneous evolution of the three mechanisms. The numerical treatment of the coupled constitutive equations is described in detail. In particular, the solution of the monolithic system of equations with the Newton-Raphson methodology, including the exact linearisation of both the local and global problems. An algorithmic strategy for handling the martensite volume fraction constraints is proposed for return-mappings in the restarted step. The model is validated using macroscopic experimental results for an austenitic stainless steel where an efficient calibration procedure for the micro-constitutive parameters is devised using derivative-free optimisation methods.

Mechanical behavior of embedded bistable dome shell with tunable energy barrier asymmetry

In this work, the mechanical response at finite strains of embedded dome shells is investigated both numerically and experimentally. These systems are interesting as they exhibit up to two stable shapes, a feature that makes them promising candidates for numerous applications from energy harvesting to shape morphing. The energy landscape scenario of these structures presents a major peculiarity as the strain energy required to switch the system between its stable shapes is path-dependent and varies upon loading (i.e. snap-through instability) and unloading (i.e. snap-back instability). This paper proposes to investigate their asymmetrical mechanical behavior. To do so, Finite Element computations were first carried out onto dome shell models, where the geometrical parameters were varied systematically. Specifically, two sets of computations were conducted using the Static Damping Method. The first served to identify the mono- and bistable domains of the embedded dome shells as function of the dome main geometrical parameters. The second focused on bistable dome geometries and explored systematically the asymmetry in the energy barrier required to switch between the two equilibrium states. Interestingly, the results of this study showed that a simple asymmetry indicator could be used to effectively qualify the dome asymmetric bistability, in turn providing simple guidelines for the design of morphing structures with programmable response. Finally, in order to validate the numerical results, the mechanical response of the 3D-printed rubber-like dome shells was measured experimentally using a dedicated set-up that was designed and fabricated to this purpose. The results of experiments were found to be in good agreement with those of simulations.

Singular travelling waves in soft viscoelastic solids of rate type

We consider shear wave propagation in soft viscoelastic solids of rate type. Based on objective stress rates, the constitutive model accounts for finite strain, incompressibility, as well as stress- and strain-rate viscoelasticity. The theory generalises the standard linear solid model to three-dimensional volume-preserving motions of large amplitude in a physically-consistent way. The nonlinear equations governing shear motion take the form of a one-dimensional hyperbolic system with relaxation. For specific objective rates of Cauchy stress (lower- and upper-convected derivatives), we study the propagation of acceleration waves and shock waves. Then we show that both smooth and discontinuous travelling wave solutions can be obtained analytically. We observe that the amplitude and velocity of steady shocks are very sensitive to variations of the stress relaxation time. Furthermore, the existence of steady shocks is conditional. Extension of these results to the case of multiple relaxation mechanisms and of the Jaumann stress rate is attempted. The analysis of simple shearing motions is more involved in these cases.

Simultaneous solution of ropes and membranes on all level sets within a bulk domain

AbstractWe formulate a mechanical model in the context of the finite strain theory for implicitly defined ropes and membranes, which applies simultaneously to all level sets of a level‐set function over a bulk domain. Further, a numerical method to solve this model is introduced. Tangential operators, which occur in the geometrically non‐linear model, are defined in the framework of the tangential differential calculus (TDC). The weak form of the governing PDE, which is required for the numerical simulation, is formulated using the co‐area formula. For the numerical analysis, the bulk domain is discretized with higher‐order finite elements which are conforming to the boundary but do not have to be aligned to the level sets within the domain. The resulting numerical method may be called Bulk Trace FEM. Higher‐order convergence studies of numerical examples, based on the residual error, the stored energy error, or the length/area of the deformed structures, ensure the validity of this method and confirm optimal convergence rates.

On the modeling of cavitation and yielding in rubber-toughened amorphous polymers: Fully implicit implementation and optimization-based calibration

In the present contribution, a finite strain visco-elastic visco-plastic (three-dimensional) constitutive model is proposed to predict the nonlinear response of neat, porous and rubber-toughened amorphous polymers. In this context, a well-known nucleation law is modified and coupled with a nucleation criterion to suppress nucleation under compression and shear loadings, as well as to identify its material parameters directly. The growth of both the nucleated and initial voids is ruled by a modified version of the Gurson’s potential, ensuring the coherence in the energy dissipation between the macro and microscales. The post-yield strain hardening of the material is modeled with the well-established eight-chain model. A fully implicit integration algorithm is formulated under the infinitesimal strains functional format, and both constitutive state update and consistent tangent modulus are properly established. A three-stage optimization-based calibration procedure is designed to identify the model’s material parameters efficiently. This decoupled calibration strategy is only possible due to the well-defined influence of the model’s material parameters in the numerical response. The predictive capability of the proposed constitutive model is validated against literature results for polycarbonate and three rubber-filled polycarbonate blends. Moreover, the numerical response of the model is assessed under different triaxial states and is further compared with the homogenized response of a Representative Volume Element of the voided-amorphous polymer microstructure. The results highlight that the model can capture the typical behavior of amorphous polymers, namely yield stress, strain-softening and strain hardening under different loading conditions, as well as the decrease of stress associated with the increase of the void volume fraction. The additional numerical analyses conducted, validated some constitutive assumptions. The efficiency of the calibration procedure and the overall numerical strategy is also demonstrated.

Damage mechanisms and analytical model of CFRP laminate under low-velocity normal and oblique impacts

The present work investigates the energy absorption performance of carbon fibre-reinforced polymer (CFRP) laminates subjected to low-velocity normal and oblique impacts. Experimental tests are conducted on laminates exposed to varying impact energies at both normal and oblique angles. The numerical model is developed to simulate the dynamic response of the CFRP. The research employs a combination of experimental and numerical methods to reveal the underlying damage mechanisms. The changes observed in the oblique impact response are explained through an analysis of the damage mechanisms. Analytical models for each energy absorption mechanism are proposed based on the principles of energy conservation and the theory of large bending deformation of anisotropic materials. The energy absorption performance of the laminates under oblique impacts, considering rebound and penetration conditions, is discussed. It is found that the delamination is the primary energy absorption mechanism of CFRP laminate under rebound conditions, while elastic deformation of fibres is the primary energy absorption mechanism under penetration conditions. Furthermore, the energy absorption performance of the laminate is found to be affected by the peak force and effective thickness of the plate, which change with the oblique angle. This paper makes a significant contribution by utilising experimental, numerical, and theoretical approaches to gain insights into the damage mechanisms of laminates subjected to oblique impacts.

Early Paleozoic oblique convergence from subduction to collision: Insights from timing and structural style of the transpressional dextral shear zone in the Qilian orogen, northern Tibet of China

Transpressional shear zones commonly occur in ancient and modern convergent plate boundaries to accommodate oblique plate convergence. The early Paleozoic Qilian orogen in northeastern Tibet records the subduction of Proto-Tethyan Ocean lithosphere and the accretion-collision of various magmatic arcs and continental terranes. This study focused on the Datong ductile shear zone, which represents the central part of the WNW-ESE−striking ductile shear zone along the northern margin of the Qilian block in the Qilian orogen. This structure bears key information about the evolution of oblique convergence during the early Paleozoic orogeny. The kinematics and timing of the Datong ductile shear zone were investigated via field-based, microstructural, and mica 40Ar/39Ar dating analyses. Mesostructural and microstructural data showed predominantly dextral strike-slip shearing within the Datong ductile shear zone. Microstructural features and quartz c-axis crystallographic preferred orientation patterns indicated that dextral ductile shearing occurred under lower-amphibolite-facies conditions (∼500−550 °C and ∼5.6 kbar) within the shear zone. Microstructures of quartz showed subgrain rotation (SGR) and grain boundary migration (GBM), suggesting dislocation creep−dominated deformation. A strain rate of 10−12 s−1 and a differential stress of 25−39 MPa were estimated by the rheological flow law and quartz paleopiezometry. Finite strain measurements indicated that all deformed rocks of the Datong ductile shear zone exhibit a weakly oblate ellipsoid near the plane strain. Kinematic vorticity (ranging 0.47−0.83) analysis suggested the coexistence of simple shear and pure shear strains within the Datong ductile shear zone, indicating a transpressional setting. Biotite and muscovite 40Ar/39Ar data showed that transpressional shearing deformation started in the Ordovician (before 453 Ma) and lasted to the Silurian (ca. 430 Ma). Our new data combined with regional geological data show that the deformation type, kinematics, and dynamics of the Datong ductile shear zone were controlled by the southward oblique subduction of the Paleo-Qilian Ocean (Proto-Tethyan Ocean) and the following oblique collision between the Qilian block and the Alxa block. The intensive transpressional deformation along the northern Qilian block may reflect strong coupling between the subducting Paleo-Qilian oceanic slab and the overriding Qilian block as well as a high degree of convergence obliquity during the ongoing early Paleozoic convergence.

Sound velocity of eclogite at high pressures and implications for detecting eclogitization in subduction zones

The increased velocity associated with decaying seismicity in the subducting oceanic crust usually has been attributed to its eclogitization. However, the degree of the eclogitization of subducting oceanic crust at depth remains unclear due to the lack of velocity data for eclogite at pressures of >3 GPa. In this study, the P- and S-wave velocities of eclogite aggregates composed of omphacite and garnet (Omp100, Omp80Grt20, and Omp50Grt50) were measured simultaneously at pressures of up to 8 GPa using ultrasonic interferometry. The velocities of the eclogite aggregates increased with rising pressure and garnet content. By determining the velocities at different pressures using the finite strain equation, the elastic moduli of the eclogites and their pressure derivatives were determined to be KS0 = 119−134 GPa, KS0′ = 5.2−5.3, G0 = 73−80 GPa, and G0′ = 1.5−1.6. The high-wave velocity and low VP/VS ratio of the eclogites, combined with the seismic tomography and seismicity distribution, jointly constrain the depth and composition of eclogitization in the subducted oceanic crust of Northeast Japan, and a new 1-D velocity structure is proposed. We also compiled and calculated the depth of the eclogitization and the garnet content of the subducting oceanic crusts in 27 typical locations around the Pacific Ocean. The depth of the eclogitization was positively correlated with the age of the subduction zone, and the garnet content was estimated to be 12%−32% in our model.

Exact solutions of burst pressure for thick-walled cylinders in power-law strain hardening steels

Exact solutions of burst pressure for defect-free, thick-walled cylindrical pressure vessels with capped ends are developed using the flow theory of plasticity in terms of the Tresca, von Mises, and Zhu-Leis yield criteria. The material is assumed to obey a power-law strain hardening rule, and the finite strain theory is adopted to describe large plastic deformation. On this basis, internal pressure is obtained as a complex polynomial function of effective strains on the inside and outside surfaces of the pressure vessel with the use of the Bernoulli numbers. At burst failure, the effective strains and stresses on the inside and outside surfaces are first determined, and then three flow solutions of burst pressure are obtained as a power series function of the diameter ratio (Do/Di), strain hardening exponent (n), and ultimate tensile stress (UTS). The power series solution is confirmed to agree well with the exact flow solution of burst pressure in a closed-form for each yield criterion, and the results show that the von Mises flow solution is an upper bound prediction, the Tresca flow solution is a lower bound prediction, and the Zhu-Leis flow solution is an intermediate prediction of burst pressure. Two sets of full-scale burst test data are then utilized to evaluate and validate the proposed flow solutions of burst pressure for both thin-walled pipes and thick-walled cylindrical pressure vessels.

Numerically efficient and robust interior-point algorithm for finite strain rate-independent crystal plasticity

The challenges of classic rate-independent crystal plasticity models regarding the determination of active slip systems and computation of solutions due to linear dependent slip systems are still a field of active research. In that regard, interior-point methods are promising approaches because a logarithmic barrier term naturally avoids the combinatorial active set search, and these methods are applicable even when the gradient of inequality constraints is linearly dependent. In this paper, a new interior-point algorithm suitable for rate-independent crystal plasticity at finite strains is developed. For that purpose, the postulate of maximum dissipation is recast as a constrained optimization problem whose optimality conditions are derived by utilizing interior-point approaches. This optimization problem is discretized in time by means of an exponential time integration of the flow rule, and the time-discrete equations are solved by a Newton scheme. The focus of this paper is the efficiency and robustness of the implementation of this approach. In particular, the effect of different line search procedures, favorable residual forms, preconditioning methods, barrier parameter updates and the consistent algorithmic tangent modulus are discussed and further elaborated. In addition, aspects regarding uniqueness and well-posedness of the algorithm are analyzed in a geometrically linearized setting. Efficiency and robustness of the developed algorithm are shown by numerical examples of an FCC crystal including finite element simulations without stabilizing hardening or rate-dependent effects. Furthermore, the results are compared to the well-established augmented Lagrangian formulation.

Numerical modeling of nonlinear deformation processes for shells of medium thickness

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.

A coupled model of transport-reaction-mechanics with trapping, Part II: Large strain analysis

A coupled finite strain chemo-transport-mechanical formulation with trapping is here proposed to extend a previous work set in the realm of small strain theory in continuum mechanics. The theory is rooted in non-equilibrium rational thermodynamics. The kinematics is based on a multiplicative decomposition of the deformation gradient to account for swelling and shrinking, thermal, elastic and inelastic contributions. Mass balance laws and balance of linear and angular momentum, as well as the laws of thermodynamics for a convecting body, are directly formulated in their material description, after specifications of some standard transformation rules between current and reference configuration. Thermodynamic restrictions are identified based on the functional dependence of the referential Helmholtz free energy density, which is chosen as the thermodynamic potential, and further subjected to a constitutive additive decomposition. Constitutive prescriptions for the chemical potentials, referential heat and mass fluxes, chemical kinetics and the generalized heat equation lead to the establishment of the governing equations. The theoretical framework is complemented by numerical simulations, highlighting the potential of the proposed formulation in multi-physics applications.

Discrete data-adaptive approximation of hyperelastic energy functions

Phenomenological constitutive modeling is prone to uncertainty and results in loss of information as data coming from experiments are not used directly in calculations. Data-driven approaches are a promising alternative to constitutive modeling. We present a new data-adaptive approach to model hyperelastic rubber-like materials at finite strains. Our proposed modeling procedure combines the advantages of phenomenological hyperelasticity with the data-driven paradigm of directly including experimental data in calculations. We suggest formulating a finite-element-like approximation of the strain energy function as a sum of basis functions expanded over a set of invariants multiplied by unknown parameters. The parameters are determined by an optimization algorithm to match measured experimental data (full-field displacements and global reaction forces) in a least-squares sense. We verify our approach and show that computation times are similar compared to those of phenomenological models. By numerical examples, we demonstrate the ability of our approach to re-identify O(10) parameters.