Deformation Gradient Research Articles

Wide area maps of building differential deformations

Abstract. The European Ground Motion Service (EGMS) uses Sentinel-1 data to monitor and measure land displacement across Europe, delivering information on ground motion phenomena. The EGMS service includes three types of products, enabling the analysis of annually updated ground motion data. The authors are working on a project aimed at computing the spatial gradient of land deformation and generating wide-area differential deformation maps. This type of maps can help identifying potentially vulnerable buildings and urban structures. The project involves developing a software pipeline to compute and convert original deformation data and creating a customized web viewer to display the results. Given the large volume of the processed datasets, with several millions of measurement points an accurate algorithm implementation is needed. The resulting information will aid in monitoring anthropogenic phenomena and their impact on building and infrastructures. This will be essential for urban management and risk mitigation planning.

Programable shape recovery properties and snap-through instabilities of viscoelastic and shape-memory NS metamaterials

The combination of smart material and mechanical metamaterial has the potential to bring novel properties and additional functionalities, which, however, has been ignored for a long time. This work numerically, experimentally, and analytically examines the negative stiffness (NS) metamaterials based on shape memory polymers (SMPs) with a focus on the tunable and temperature-dependent properties induced by the interaction of material and geometry nonlinearity. A universal discrete model of a series of shape-memory NS metamaterial is developed by separating the bending effect and stretching effect, where the viscoelastic material is described by the generalized Maxwell-Wiechert model, and an SMP constitutive model is proposed based on multiplicative decomposition of the deformation gradient. The mechanical and shape-memory properties influenced by temperature, geometry, material and loading rate are experimentally observed through relaxation tests and cyclic compression tests. Then the discrete model and finite element analysis (FEA) is adopted to examine the novel mechanical responses in a wide range of parameters. Results show that viscoelasticity and loading rate play a significant role in mechanical responses, and tunable load-capacity and stability can be achieved for shape-memory metamaterial. A phase diagram of different stabilities is constructed showing strictly monotonic, S-shaped, and snapping responses. Additionally, repeated compression and recovery performance proves the possibility of reusable applications. This work provides new opportunities for functional metamaterial to obtain programmable shape recovery and mechanical responses.

Analysis of shear bands in bulk metallic glasses based on elastoplastic phase transformation theory

Abstract Shear bands in amorphous alloys have been widely observed in uniaxial tension and compression experiments, reflecting strain localization, with plastic deformation occurring within the shear bands. In many instances, failures of bulk metallic glasses (BMGs) occur along the dominant shear bands. In this study, phase transformation theory is applied to investigate the mechanical properties of shear bands, focusing on the analysis of the two-phase deformations. The shear bands and the regions outside them are treated as two distinct phases in equilibrium. The deformation gradient tensor across the interface between these phases is discontinuous. By applying the jump conditions, governing equations are derived. As a case study, the shear bands and surrounding regions of plastic BMGs under uniaxial compression are analyzed, enabling the calculation of the stress associated with phase transformation and the inclination angle of the shear bands. The results obtained from this theoretical model align well with existing experimental and simulation data.

Comparative study of microstructural evolution and mechanical properties in friction stir processed, reinforced friction stir processed, and heat-treated reinforced friction stir processed Al composites

The development of friction stir welding (FSW) created the basic foundation for friction stir processing (FSP). This process entails refining the local microstructure and achieving the localised plastic deformation due to the stirring effect of the tool, which may involve adding additional filler material as required. The rotating tool involved in the process has a shoulder and pin responsible for heat generation for plastic deformation and localised mixing. The rotating tool is inserted and traversed on the desired surface, tailoring the material properties. The tool’s exposure to the surface causes localised plastic deformation and thermal gradient, and microstructural changes occur locally. The present investigation uses Friction Stir Processing (FSP) to fabricate an aluminium metal matrix composite. Recently, the use of aluminium alloy has been replaced by composites because of their increased mechanical and physical properties (specific properties). The experiment leads to manufacturing an aluminium metal matrix composite using aluminium oxide, boron carbide and commercially pure copper dust. Later, the effect of heat treatment on the respective metal matrix composite is studied. A comparative study among three different types of FSP techniques viz. FSP, Reinforced FSP, and Heat-treated Reinforced FSP have been performed. The tool revolution speed (RPM) and tool traverse speed of 400 RPM and 10 mm/min had been implemented, respectively. The T4 Heat-Treatment cycle was used during the current investigation. This investigative study was performed to analyse the changes in grain sizes and hardness values at the stir zone (SZ), thermo-mechanically affected zone (TMAZ), heat affected zone (HAZ) and on the base materials (BM) of the samples. The reinforced FSP metal matrix composite (MMC) showed an 81.26% increase in the micro-hardness value (HV) compared to the base metal. Further on analysing the microstructures, the reinforced FSP composite showed a 65.20% reduction in grain size. Moreover, the Heat-Treated sample showed an increase in the micro-hardness value by 99.8%.

Minimal surface-guided higher-order mesh generation for CAD models

This paper presents a novel method for generating higher-order meshes for CAD surfaces by leveraging minimal surface theory to improve element shapes. We explore the concept of higher-order mesh distortion through deformation gradients and introduce an energy function designed to minimize the surface area of these meshes, providing a theoretical justification for its effectiveness in untangling. The process of mesh generation starts with segmenting CAD surfaces into linear elements, followed by the insertion of higher-order nodes within these elements. These nodes are then projected onto the CAD surface to form the initial higher-order elements. By optimizing energy functions related to minimal surfaces and the projection distances, we achieve high-quality, geometrically accurate higher-order surface meshes. Our method has been validated on complex geometries, showcasing its potential in creating effective higher-order meshes for industrial CAD models.

Bending-neutral deformations of minimal surfaces

Minimal surfaces are ubiquitous in nature. Here, they are considered as geometric objects that bear a deformation content. By refining the resolution of the surface deformation gradient afforded by the polar decomposition theorem, we identify a bending content and a class of deformations that leave it unchanged. These are the bending-neutral deformations, fully characterized by an integrability condition; they preserve normals. We prove that: (i) every minimal surface is transformed into a minimal surface by a bending-neutral deformation; (ii) given two minimal surfaces with the same system of normals, there is a bending-neutral deformation that maps one into the other; and (iii) all minimal surfaces have indeed a universal bending content.

Modeling and Simulation of the Aging Behavior of a Zinc Die Casting Alloy

While zinc die-casting alloy Zamak is widely used in vehicles and machines, its solidified state has yet to be thoroughly investigated experimentally or mathematically modeled. The material behavior is characterized by temperature and rate sensitivity, aging, and long-term influences under external loads. Thus, we model the thermo-mechanical behavior of Zamak in the solid state for a temperature range from −40 °C to 85 °C, and the aging state up to one year. The finite strain thermo-viscoplasticity model is derived from an extensive experimental campaign. This campaign involved tension, compression, and torsion tests at various temperatures and aging states. Furthermore, the thermo-physical properties of temperature- and aging-dependent heat capacity and heat conductivity are considered. One significant challenge is related to the multiplicative decompositions of the deformation gradient, which affects strain and stress measures relative to different intermediate configurations. The entire model is implemented into an implicit finite element program and validation examples at more complex parts are provided so that the predicability for complex parts is available, which has not been possible so far. Validation experiments using digital image correlation confirm the accuracy of the thermo-mechanically consistent constitutive equations for complex geometrical shapes. Moroever, validation measures are introduced and applied for a complex geometrical shape of a zinc die casting specimen. This provides a measure of the deformation state for complex components under real operating conditions.

Mini-Satellite Fucheng 1 SAR: Interferometry to Monitor Mining-Induced Subsidence and Comparative Analysis with Sentinel-1

Mining-induced subsidence poses a serious hazard to the surrounding environment and infrastructure, necessitating the detection of such subsidence for effective disaster mitigation and the safeguarding of local residents. Fucheng 1 is the first high-resolution mini-satellite interferometric Synthetic Aperture Radar (SAR) launched by China in June 2023. In this study, we used Fucheng 1 SAR images to analyze mining-induced subsidence in Karamay by InSAR Stacking and D-InSAR. The findings were compared with Sentinel-1A imagery to evaluate the effectiveness of Fucheng 1 in monitoring subsidence and its interferometric performance. Analysis revealed significant mining-induced subsidence in Karamay, and the results from Fucheng 1 closely corresponded with those from Sentinel-1A, particularly regarding the extent of the subsidence. It is indicated that the precision of Fucheng 1 SAR imagery has reached leading standards. In addition, due to its higher resolution, the maximum detectable deformation gradient (MDDG) of Fucheng 1 is 2.15 times higher than that of Sentinel images. This study provides data support for the monitoring of mining-induced subsidence in the Karamay and give a theoretical basis for the application of Fucheng 1 in the field of Geohazard monitoring.

Transformation of surface [formula omitted]-forces in high grade elasticity

The expression of the elastic energy, postulated for a given material, specifies through the principle of virtual work which external loadings turn out to be admissible. For the elastic bodies, the energy of which depends on the gradients of the placement higher than (H+1) (or, equivalently, on the derivatives of the deformation gradient higher than H), external force densities can be prescribed over the boundary surface, which do work on the H-th derivative of the virtual displacements along the normal direction. Such generalized forces with H=1,2 have been dealt with in the literature, referred to as double and triple forces: when H is increased, however, the analytical evaluation of these work terms may become prohibitive. Each Eulerian H-force density of this kind, acting on the current boundary surface, turns out to be equivalent to a set of diverse actions defined in the Lagrangian configuration. These actions include: (i) surface terms doing work on the virtual displacement vector; (ii) surface terms doing work on the higher order derivatives of the virtual displacement along the normal direction; (iii) edge terms, doing work on the virtual displacement or on its higher order gradients, which may be further reduced. In this study, a recursive algorithm is utilized to generate all these Lagrangian contributions for any H. Such an approach rests on the binomial expansion of the tensor products of complementary surface projectors, symmetrized in the work duality, and on the integration by parts combined with the generalized divergence theorem.

Phenomenological constitutive laws for the dissipative behaviour of highly compressible elastomers and their finite element implementation

This paper deals with the development of constitutive equations to model the mechanical behaviour of compressible elastomers. These materials are naturally incompressible but can be made compressible by the addition of hollow microspheres, for example. Such a material is referred to as syntactic foam. The CEA (French Commission for Atomic and Alternative Energies) employs such compressible materials as seals in complex structures to reduce the internal stresses in vulnerable components and prevent their failure. The behaviour of these structures is predicted by finite element simulations. It is important to know and model the mechanical behaviour of the seals. Like elastomers, they can undergo large deformations. The microspheres enable the material to undergo large volume change unlike pure elastomer that is nearly incompressible. This compressibility also intensifies dissipative phenomena encountered in elastomers such as viscosity or plasticity. Furthermore, the Mullins stress softening effect is also intensified even for loadings that only bring about volumetric changes. To model these behaviours, a phenomenological approach was developed based on the isochoric/volumetric decomposition of the deformation gradient. The method of intermediate dissipative configurations was employed to introduce multiple phenomena, including viscosity (with several characteristic times) and viscoplasticity, for these two parts of the deformation. The constitutive equations and their flow rules were implemented in Abaqus through a UMAT subroutine and using a numerical approach to define the tangent operator. The parameters of the behaviour law were identified using a model reduction technique known as shape manifold approach. The resulting model can be compared with experimental data.

A covariant formulation for finite strain modelling of orthotropic elasticity and orthotropic plasticity with plasticity-induced evolution of orthotropy: Application to natural fibres

We introduce a rate-independent model for orthotropic elastic and orthotropic plastic material behaviour in a hyper-elasto-plastic framework at finite strains. The model is based on the postulate of covariance and does not rely on a multiplicative decomposition of the deformation gradient. Furthermore, a plastic-deformation-induced evolution of orthotropy is considered, similar to the notion of plastic spin. We propose that the orthotropic strain energy function and the orthotropic yield criterion are guided by identical structural tensors that evolve with plasticity. The modelled material behaviour is significant for natural fibres such as flax, hemp, or pulp fibres. Our formulation has three findings. Firstly, the covariant formulation of plasticity provides rate equations for the plastic variables and the structural tensors suitable for reproducing stress–strain diagrams of natural fibres. Secondly, the introduction of plastic-deformation-induced evolution of orthotropy in the proposed covariant setting results in a non-associative plasticity algorithm. Thirdly, the covariant setting allows the incorporation of suitable constitutive equations for the structural tensors to evolve orthotropy. The latter successfully models the stiffness increase in stress–strain diagrams of cyclic tensile tests of natural fibres.

A finite strain viscoelastic model with damage and tension–compression asymmetry considerations for solid propellants

A short survey on the experimental testing of solid propellants has highlighted finite strain responses that are temperature-dependent, viscoelastic with damage, and exhibit tension/compression asymmetry. Consequently, a finite strain viscoelastic model that satisfies the principles of thermodynamics has been developed. This model is based on the common multiplicative decomposition of the deformation gradient into elastic and viscous components, with considerations for damage and asymmetry. The model has been tested against three sets of data from the literature, carefully selected to represent the various characteristics of solid propellants. The model accurately reproduces uniaxial tension responses at different strain rates and temperatures, with the capability to account for superimposed hydrostatic pressure. Notably, these satisfactory representations require only five fitting parameters, in addition to the typical identification of polymer linear viscoelasticity and time–temperature superposition. Finally, an attempt to reproduce both tension and compression tests conducted independently on the same material underscores the need to account for tension–compression asymmetry, as defined in the proposed constitutive equations. This finding advocates for new tests, such as compression following tension and vice versa.

Isotropic growth model for generalized scaled boundary isogeometric analysis on slender structures

AbstractComponents for timber constructions are usually based on simple geometrical shapes. In the production, bifurcated or curved components of the tree crown remain unused. Recently, building botany has evolved as a new discipline of living construction design. Therefore, trees are lead in the right shape to join as a load‐bearing structure interconnected with conventional parts. For the structural analysis, geometric features such as curvature or bifurcations exhibit unique static capabilities. It is crucial to transfer these geometries into the calculation model accurately. The structure can be depicted through three‐dimensional scanning techniques. A fundamental model (point cloud) is created in this process. This is the base for a Non‐uniform rational B‐spline (NURBS) model, existing as a 2D surface within a 3D space. Using the novel generalized scaled boundary isogeometric analysis (GSBIGA)‐approach, numerical disadvantages of conventional scaled boundary methods can be reduced. This method employs a scaling center line instead of a scaling center point. A growth simulation is used to describe the leading process. The present approach is based on an isotropic growth model applied to wooden growth applications. Therefore, the deformation gradient is split multiplicatively into an elastic and a growth part. The formulation for modeling volumetric growth is located on the kinematic level.

Deformation and microstructure radial gradient evolution of AZ31 magnesium alloy bar during three-roll skew rolling

AZ31 magnesium alloy bar with radial gradient microstructure was prepared by three-roll skew rolling process (TRS), and achieving an excellent combination of ductility and strength. To elucidate the formation mechanism of this gradient microstructure, the deformation behavior during rolling was analyzed through experiments and simulations. The simulation results indicated that the metal flow of the bar in a spatial helical trajectory during rolling, with the trajectory inclination angle gradually decreasing from the center to the surface, promoting dynamic recrystallization in the surface and near-surface regions. The radial equivalent strain decreased from 1.5 at the surface to 0.01 at the center, and the shear strain decreased from 0.051 at the surface to 0.00012 at the center. Experimental results showed that the grain size of the bars decreased from 49.77 µm at the center to 19.72 µm at the surface, and the tensile twin volume fraction decreased from 34.1 % to 19.4 %. Dynamic recrystallization predominantly occurred in the outer ring region of the bars, while significant {10−12} tensile twin activation was observed in the central region, with twin-induced dynamic recrystallization observed at the twin boundaries. Mechanical property tests demonstrated that the gradient structure endowed the bars with a favorable combination of strength and elongation, with hardness significantly increasing from 50 HV at the center to 90 HV at the surface. This indicates that the three-roll skew rolling technology effectively promotes dynamic recrystallization at the surface, and the resulting gradient microstructure achieves an excellent combination of strength and plasticity in magnesium alloy bars.

Prediction of sectional collapse of thin-walled structure under pure bending by nonlinear composite beam theory

Brazier (1927) found that when one dimension of the beam cross-section was relatively smaller than the others, large in-plane displacements over the cross-section might occur, even though the strains could remain very small. Under this circumstance, the so-called Brazier effect refers to the cross-sectional ovalization, which leads to nonlinear bending buckling and collapses. This paper studies the Brazier effect by the nonlinear Variational Asymptotic Beam Sectional Analysis (VABS) theory which considers finite cross-sectional deformations. Nonlinear VABS reduces three-dimensional (3D) continuum to a one-dimensional (1D) beam analysis and a two-dimensional (2D) cross-sectional analysis featuring both geometric and material nonlinearities without unnecessary kinematic assumptions. The present theory is implemented using the finite element method (FEM) in the VABS code, a general-purpose beam cross-sectional analysis tool. An iterative method is applied to solve the finite warping field for the classical-type model using the Euler–Bernoulli beam theory. The deformation gradient tensor is directly used to deal with finite deformation, various strain definitions, and several types of material laws. Numerical examples demonstrate the capabilities of VABS to predict the sectional collapse of thin-walled structures under pure bending.

On effective surface elastic moduli for microstructured strongly anisotropic coatings

The determination of surface elastic moduli is discussed in the context of a recently proposed strongly anisotropic surface elasticity model. The aim of the model was to describe deformations of solids with thin elastic coatings associated with so-called hyperbolic metasurfaces. These metasurfaces can exhibit a quite unusual behaviour and concurrently a very promising wave propagation behaviour. In the model of strongly anisotropic surface elasticity, strain energy as a function of the first and second deformation gradients has been introduced in addition to the constitutive relations in the bulk. In order to obtain values of surface elastic moduli, we compare dispersion relations for anti-plane surface waves obtained using the two-dimensional (2D) model and three-dimensional (3D) straightforward calculations for microstructured coatings of finite thickness. We show that with derived effective surface moduli, the 2D model can correctly describe the wave propagation.

Calculation of 3D Shape of a Hyperelastic Body for Nonlinear Elasticity Models Using the Newton Method

This article discusses methods for calculating the deformations of objects with hyperelastic materials within the framework of the nonlinear elasticity theory. This topic is relevant due to the use of new technological materials in industry, and as a result, the emerging task of preliminary numerical calculations for operational reliability, research and modeling of deformation behavior. The first part provides a brief overview of the main provisions and formulas of the nonlinear elasticity theory. Then the spatial discretization of 3-dimensional objects and the calculation of the deformation gradient using the finite element method are considered. The article provides an algorithm for solving this problem, such as minimizing the functional of stored energy, and also considers the class of permissible deformations. Afterwards, a detailed description is given, along with pseudocode, of the method of implementing and calculating the minimization by Newton’s method. The last part demonstrates an example of the calculations performed, based on the developed software that allows for numerical experiments and computer modeling of deformations of hyperelastic bodies, and one of the capabilities of which is to perform these calculations using Newton’s method.

A comparison of finite strain viscoelastic models based on the multiplicative decomposition

The constitutive equations of several finite strain viscoelastic models, based on the multiplicative decomposition of the deformation gradient tensor and formulated in a thermodynamically consistent framework, are reviewed to demonstrate their similarities and differences. The proposed analysis shows that dissipation formulations, which may appear different, are similar when expressed in the same configuration, enabling the definition of a unified general model. The ability of this general model to reproduce the main features of the behavior of rubbers is then explored. First, its responses are compared to those of finite linear viscoelastic models commonly implemented in commercial finite element codes. Cases of monotonic uniaxial tension and simple shear, relaxation, and sinusoidal simple shear are considered. Second, a comparison is made between a classic generalized Maxwell rheological scheme and a Zener one with a non-constant viscosity, exploring the relevance of both options within the general model’s constitutive equations.

Static and dynamic stabilities of modified gradient elastic Kirchhoff–Love plates

The static and dynamic stabilities of modified gradient elastic Kirchhoff–Love plates (MGEKLPs), which incorporate two length-scale parameters related to strain gradient and rotation gradient effects, are comprehensively analyzed under various load forms and boundary conditions (BCs). The study of static stability employs static balance method and an improved energy method by introducing higher-order deformation gradients and corresponding energy terms. Utilizing the variational method, a sixth-order fundamental buckling differential equation for MGEKLPs under both transverse and in-plane loads is derived, serving as the foundation for the static balance method. The static stability analysis of MGEKLPs examines the combined effects of strain and rotation gradients on size-dependent critical buckling loads. Building on generalized strain energy with higher-order deformation energy, the energy method of classical elastic thin plate model is enhanced and applied to the static stability analysis of MGEKLPs. This approach enables the investigation of static stability without being constrained by the need to solve complex differential equations, making it applicable to various BCs and load scenarios. While static stability provides a description of stable state of an elastic system, dynamic stability offers a more scientific and rigorous analysis. The dynamic stability of simplified gradient elastic Kirchhoff–Love plates (SGEKLPs) with curved edges and different BCs is further investigated by combining the generalized strain energy with Lyapunov’s second stability method, presenting the dynamic stability criterion in the form of norms. A strict description of the dynamic stability of a SGEKLP over the entire time domain is provided for different supporting conditions, including case where all edges are supported and case with free edges. The analysis of size-dependent static and dynamic stabilities offers theoretical guidance for designing elastic thin plates with microstructures.

A method to represent the strain hardening effect in the hyper-elastic model within a fully Eulerian framework

We propose a method to treat the strain hardening effect in the hyperelastic model. Coupled with the level set interface capturing method and a real ghost fluid method, it is formulated in a fully Eulerian framework. The HLLD approximate Riemann solver is used to obtain the interface state. We use the elastic inverse deformation gradient tensor to track the material deformation state. Once the solid starts to yield, along the normal direction in stress space, an iterative algorithm is used to relax the elastic inverse deformation gradient tensor to target value where the equivalent stress meets the requirement of consistency condition in plasticity. This approach is intuitive and can avoid the use of the physical delay time, which is difficult to calibrate from experiments. We use the Wilkins flying plate impact, cylindrical shell collapse, Taylor impact and the conical-nosed projectile penetration test cases to validate the method. The results show good agreement with the theoretical analysis and experiments. This approach can also apply without further extension when thermal softening effect or damage models are considered.