Deformation Gradient Research Articles

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.

Multi-layered solid element for seismic analysis of rubber bearings considering stress continuity based on finite particle method

Rubber bearings are crucial components in seismic isolation systems to mitigate the damaging effects of earthquakes on structures. Rubber bearings generally consist of alternate layers of steel and rubber bonded together. Simplified two-point spring models omit this layered construction and have limitations in adaptability and programmability, while refined multiple solid models lead to substantial computational costs. This study presents an effective and efficient approach to simulate rubber bearings by using multi-layered solid element based on the finite particle method (FPM). The traditional “Zigzag” shape function is extended into the 3D situation to reflect the serrated deformation within the layered construction. Based on the simplified assumption of stress continuity, an iterative approach is presented for the deformation gradient to achieve the force continuity between layers in dynamic nonlinear scenarios. The multi-layered solid models for both natural rubber bearings (NRBs) and lead rubber bearings (LRBs) are proposed based on this element. The errors of the hysteresis curves of rubber bearings are below 3.1 % between the numerical results obtained by using the multi-layered solid model and the experimental results. The computational speedup ratio of the proposed multi-layered solid model is verified to be 19 relative to the refined multiple solid model. The proposed method is applied to a base-isolated frame structure to demonstrate its effectiveness in structural seismic analysis.

A large deformation viscoelasticity theory for elastomeric materials and its numerical implementation in the open-source finite element program FEniCSx

Elastomeric solid materials typically exhibit a pronounced viscoelastic response. In this paper we consider a large deformation viscoelasticity theory for isotropic elastomeric materials which uses a multi-branch multiplicative decomposition of the deformation gradient. We then describe the numerical implementation of the theory in the open-source finite element program FEniCSx. Several example simulations which demonstrate the capability of the theory and its numerical implementation to model stress-relaxation, creep, stretch-rate sensitivity, hysteresis, damped inertial oscillations, and dynamic column buckling are shown. The source codes for these simulations are provided. The theory and the codes presented in this paper lay the foundation for future extensions of the theory and its numerical implementation to include the effects of coupling with thermal, electrical, and magnetic fields — extensions which are of central importance in modeling the response of soft-active materials.

A continuum model for the mechanics of elastomeric sheets reinforced with extensible bidirectional fibers resistant to lateral pressure

We investigate the concurrent three-dimensional (in-plane and out-of-plane) deformations of fiber-reinforced composite (FRC) sheets undergoing lateral pressure. This involves the utilization of the Neo-Hookean strain energy model for the matrix material and computing the strain energy of bidirectional fibers by accounting for the stretching, bending, and twisting responses of the fibers. The kinematics of FRC are formulated within the framework of differential geometry on FRC surfaces, including the computations of the first and second gradient of deformation. By employing the variational principles, we derive the Euler equations describing the mechanics of the fiber–matrix composite system subjected to lateral pressure. The resulting three-dimensional continuum model theoretically predicts the deformation of the matrix material and it is found that the maximum deformation of matrix material occurs in the diagonal direction of the domain, yet, the center of the domain suffers weak in-plane deformation because of surface tension equilibrium. In addition, the stretching, bending, and twisting kinematics of fiber units are computed to investigate the effects of the individual fiber’s kinematics on the overall deformation of fiber meshwork. Lastly, it is found that the lateral pressure on the FRC surface induces fiber flexure in the vicinity of domain boundaries and fiber stretch inside the domain, corresponding to the intensified shrinking strain near the edges and stretching strain in the middle of the domain. The theoretical results provide phenomenologically meaningful insights into comprehending the damage patterns of the fiber-reinforced building material, the hemispherical dome shaping results of bamboo Poly (lactic) acid (PLA) composites and the out-of-plane deformation of woven fabric.

Effect of CSET processing on a commercial purity polycrystalline Al sample and numerical simulation of induced plastic deformation

A commercial purity Al sample was processed using complex shearing of the extruded tube (CSET) technique, which enables the preparation of tubes from bulk samples in one step by combining extrusion and two Equal Chanel Angular Pressing - like passes. In the present paper, the CSET process was simulated and analysed using Finite Element Method. The results of the simulations were compared with the experimental findings. Both modelling and experiment revealed a gradual increase in plastic deformation in the material during CSET processing. Simultaneously, microscopic investigations confirmed concurrent grain refinement. Moreover, a slight gradient of plastic deformation predicted by FEM in the main deformation zone was confirmed by microhardness measurements. In the resulting tube wall, electron backscattered diffraction experiments revealed the presence of fine grains and a partially recovered microstructure. The Vickers microhardness was found to be doubled by the CSET processing.

Конечно-элементное моделирование нелинейных задач упругости в абсолютных узловых координатах на неструктурированных шестигранных сетках

We consider a finite element approach to solving the problem of elasticity theory in terms of absolute nodal coordinate formulation (ANCF), in which large body displacements are described in a global reference frame without using any local coordinate system. The main feature of the method is the absence of gyroscopic effects and, as a result, constancy of the mass matrix and the vector of the generalized force of gravity. In contrast to the traditional ANCF approach, the sets of nodal degrees of freedom of the finite element are formed only on the basis of absolute coordinates of nodes, which allows us to solve the problem using, in particular, unstructured hexahedral meshes. To construct the stiffness matrix, we apply a second-order automatic differentiation algorithm, which ensures its symmetrical form (Hessian matrix) and is analytically accurate in calculating the derivative. This approach also makes it possible to carry out calculations for models of hyperelastic materials without the corresponding Piola-Kirchhoff tensor. It has been shown that within the framework of discretization of the equation of motion, along with the well-known Newmark numerical integration scheme, it is possible to use the HTT-α scheme, which is unconditionally stable, second-order accurate and dissipative for high frequencies. We present several examples of solving static and dynamic elasticity problems for compressible and incompressible models of hyperelastic materials, in which the functions of internal energy density of the body are specified in terms of the deformation gradient.

Controlled Interlayer Binding and Healing Improve the Transverse Properties of Upcycled Disposable Waste Wood.

Recovery and reuse of bulk waste wood are particularly challenging because of usage defects and contaminations. Here, we present a robust and efficient strategy for regenerating used wood veneers into high-performance structural materials through micro/nano interface manipulation. Our approach involves using cellulose-based interlayers to bind together two waste wood plates without an external adhesive by partially dissolving and regenerating the interlayer using a solution of ionic liquids and dimethyl sulfoxide. The mechanical properties of the regenerated wood exceed that of natural wood, displaying over a 16 and 20 times increase in transverse tensile strength and modulus, respectively, and 4-6 times improvement in longitudinal tensile strength and modulus. Nanoscale mechanical analyses show that the improvement is possible as a result of several factors, including the robust network structure of the interlayer, the good adhesion at the wood-interlayer interface, the compacted wood structure, and the low stiffness and deformation gradients between the interlayer and the wood structure. The interlayers can be created from waste papers and wood particles by taking advantage of the nanofibrillar structure of cellulose.

Approximation‐based implicit integration algorithm for the Simo‐Miehe model of finite‐strain inelasticity

AbstractWe propose a simple, efficient, and reliable procedure for implicit time stepping, regarding a special case of the viscoplasticity model proposed by Simo and Miehe (1992). The kinematics of this popular model is based on the multiplicative decomposition of the deformation gradient tensor, allowing for a combination of Newtonian viscosity and arbitrary isotropic hyperelasticity. The algorithm is based on approximation of precomputed solutions. Both Lagrangian and Eulerian versions of the algorithm with equivalent properties are available. The proposed numerical scheme is non‐iterative, unconditionally stable, and first order accurate. Moreover, the integration algorithm strictly preserves the inelastic incompressibility constraint, symmetry, positive definiteness, and w‐invariance. The accuracy of stress calculations is verified in a series of numerical tests, including non‐proportional loading and large strain increments. In terms of stress calculation accuracy, the proposed algorithm is equivalent to the implicit Euler method with strict inelastic incompressibility. The algorithm is implemented into MSC.MARC and a demonstration initial‐boundary value problem is solved.

A bond-augmented stabilized method for numerical oscillations in non-ordinary state-based peridynamics

Non-ordinary state-based peridynamics has been widely used due to its capability to combine non-local theory with conventional material models. However, it suffers from numerical oscillations in calculations. The fundamental reason is that the approximate deformation gradient causes the non-unique mapping between deformation states and force states. To address this issue, the present work utilizes the penalty function method to reformulate the deformation gradient for each individual bond. The force state corresponding to a single bond is then constructed based on this newly developed bond-augmented deformation gradient in the framework of least action principle. This bond-augmented force state has a strong dependence on the deformation of that particular bond, thus well resolving the non-unique mapping issue. The resulting bond-augmented stabilized method does not introduce an additional local bond-associated horizon or extra force state termed as that in the previous studies and does not require tedious parameter adjustments. Numerical results show that the proposed bond-augmented stabilized method can significantly remove numerical oscillations and have good accuracy and convergence. Moreover, the proposed bond-augmented stabilized method is also proved to readily capture crack paths.

A Particle Based Approach For Improved Resolution PIV And TOMO-PIV Based On The Coherent Point Drift And The Affine Least-Squares Transformation

Particle Image Velocimetry (PIV) is a widely used method for flow diagnostics, but there is still potential for improvement, particularly in terms of velocity gradient estimation and computational cost when considering three-dimensional problems. This paper presents a framework that combines a Particle Tracking Velocity (PTV) approach with local gradient-based Eulerian reconstruction to improve PIV performance. The approach uses the Coherent Point Drift (CPD) method for particle pairing and introduces the Affine Least-Squares Transformation (ALST) for local deformation gradient estimation. The CPD method consists of pairing particles whose positions at two successive instants have been obtained from the images. The ALST estimates local deformation gradients, allowing the Eularian reconstruction of the velocity field. The effectiveness of the proposed method compared to traditional PIV algorithms is demonstrated by synthetic test cases in both 2D and 3D configurations. In 2D cases, the CPD+ALST approach outperforms standard PIV methods, especially in capturing local velocity gradients. In 3D cases, comparisons with TOMO-PIV show the improved performance of CPD+ALST in the context of locally large velocity gradients. However, the linear nature of ALST makes it less efficient than quadratic binning techniques. The study demonstrates the potential of CPD+ALST to improve velocity field reconstruction in complex flows.

Multiscale simulation of spatially correlated microstructure via a latent space representation

When deformation gradients act on the scale of the microstructure of a part due to geometry and loading, spatial correlations and finite-size effects in simulation cells cannot be neglected. We propose a multiscale method that accounts for these effects using a variational autoencoder to encode the structure–property map of the stochastic volume elements making up the statistical description of the part. In this paradigm the autoencoder can be used to directly encode the microstructure or, alternatively, its latent space can be sampled to provide likely realizations. We demonstrate the method on three examples using the common additively manufactured material AlSi10Mg in: (a) a comparison with direct numerical simulation of the part microstructure, (b) a push forward of microstructural uncertainty to performance quantities of interest, and (c) a simulation of functional gradation of a part with stochastic microstructure.

Total Lagrangian smoothed particle hydrodynamics with an improved bond-based deformation gradient for large strain solid dynamics

Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH) is one variant of SPH where the variables are described using the fixed reference configuration and a Lagrangian smoothing kernel. TLSPH elevates the computational efficiency of the standard SPH when no topological change is involved, and it alleviates the stability of SPH scheme with respect to tensile loading. However, instabilities associated with spurious mode, or hourglass/zero-energy mode, persists and often affects the simulation of solids undergoing extremely large strain. This work proposes an alternative formulation to compute deformation gradient with improved accuracy and therefore minimising the possibility of encountering the zero-energy mode. Specifically, we leverage the local discrete computation of bond-based (or pairwise) deformation gradient smoothed by the kernel. Additionally, the bond of a particle with itself is considered to preserve the polynomial reproducibility imposed by the kernel correction scheme. We showcase the convergence of the approach using a two-dimensional benchmark example. Furthermore, the accuracy, robustness, and stability of the proposed approach are assessed in various two- and three-dimensional examples, highlighting on the stability improvement that allows for solid dynamic simulations with more extreme elongation.

Visco-Hyperelastic Constitutive Modeling for High-Damping Rubber Materials During Combined Quasi-Static Compression–Cyclic Shear Deformation Process

High-damping rubber materials utilized in high-damping rubber isolation bearings are frequently subjected to multiple deformations during the occurrence of earthquakes. Typically, large combined compression–shear deformations of the material could potentially cause compressive shear damage to rubber bearings. During this process, visco-hyperelastic properties of rubber materials will greatly change, which would significantly impact the seismic performance of rubber bearings. Thus, to give out a deep insight into their variations, it is necessary and urgent to develop a high-performance numerical method to investigate this process. This paper proposed a visco-hyperelastic constitutive modeling approach for high-damping rubber materials based on the experimental assessment of combined quasi-static compression–cyclic shear deformation process. Within the thermodynamic framework, the Clausius–Duhem inequality associated with the intrinsic dissipation of the material was firstly derived in accordance with the Lagrangian formulism. Then, stress–strain relations were obtained upon considering the occurrence of entropy production due to viscous dissipation. In the model, Stumpf–Marczak strain energy density function, which satisfies the Baker–Ericksen (B–E) inequality, was harnessed to describe the hyperelasticity of the material. By introducing higher orders of strain and strain rates and taking their couplings into account, a generalized viscous dissipation potential was proposed to capture nonlinear strain and strain rate-sensitivity effects of the material. To identify constitutive parameters, the deformation gradient was particularized for the combined quasi-static compression–cyclic shear deformation process. And, an inverse identification procedure was carried out at different levels of compression stress. The prediction results revealed that the proposed model exhibits remarkable prediction ability and adaptivity for different rubber materials during this process. Several new insights were highlighted on the variations of visco-hyperelastic characteristics of high-damping rubber materials with respect to the compression stress. The accuracy of the model was further validated by design parameters including initial shear modulus, secant shear modulus and equivalent viscous damping factor. This work could provide a fundamental guideline for the optimization and reliability analysis of high-damping rubber isolation bearings used in the field of seismic engineering.

Nonlinearly elastic maps: Energy minimizing configurations of membranes on prescribed surfaces

We propose a model for nonlinearly elastic membranes undergoing finite deformations while confined to a regular frictionless surface in R 3 \mathbb {R}^3 . This is a physically correct model of the analogy sometimes given to motivate harmonic maps between manifolds. The proposed energy density function is convex in the strain pair comprising the deformation gradient and the local area ratio. If the target surface is a plane, the problem reduces to 2-dimensional, polyconvex nonlinear elasticity addressed by J. M. Ball. On the other hand, the energy density is not rank-one convex for unconstrained deformations into R 3 \mathbb {R}^3 . We show that the problem admits an energy-minimizing configuration when constrained to lie on the given surface. For a class of Dirichlet problems, we demonstrate that the minimizing deformation is a homeomorphism onto its image on the given surface and establish the weak Eulerian form of the equilibrium equations.

A generalized peridynamic material correspondence formulation using non-spherical influence functions

Peridynamics theory is a nonlocal continuum mechanics theory. The peridynamic material correspondence formulation provides passage for direct incorporation of any material constitutive models from the classical local continuum mechanics theory. The performance of the material correspondence formulation heavily relies on the influence function whose currently used form is spherical and depends only on the bond length. Together with the way how the nonlocal deformation gradient is constructed based on the material point horizon, this results in the well-known issue of material instability in the conventional peridynamic material correspondence formulation. In this paper, a generalized peridynamic material correspondence formulation that enables use both spherical and non-spherical influence functions is developed. A recently proposed class of parameterized non-spherical influence functions that depend on both the bond length and bond direction is used in the generalized formulation. Numerical examples including wave dispersion, small and finite deformation and linear elastic fracture analyses are studied to check the material stability, convergence characteristic, prediction accuracy and applicability to fracture problems without singularity issue of the proposed formulation. It is found that the proposed formulation is inherently stable, possesses linear convergence rate, yields highly accurate predictions for both small and finite deformation problems, and small horizon can be used to improve computational efficiency.

Imposing Dirichlet boundary conditions directly for FFT-based computational micromechanics

We discuss how Dirichlet boundary conditions can be directly imposed for the Moulinec–Suquet discretization on the boundary of rectangular domains in iterative schemes based on the fast Fourier transform (FFT) and computational homogenization problems in mechanics. Classically, computational homogenization methods based on the fast Fourier transform work with periodic boundary conditions. There are applications, however, when Dirichlet (or Neumann) boundary conditions are required. For thermal homogenization problems, it is straightforward to impose such boundary conditions by using discrete sine (and cosine) transforms instead of the FFT. This approach, however, is not readily extended to mechanical problems due to the appearance of mixed derivatives in the Lamé operator of elasticity. Thus, Dirichlet boundary conditions are typically imposed either by using Lagrange multipliers or a “buffer zone” with a high stiffness. Both strategies lead to formulations which do not share the computational advantages of the original FFT-based schemes. The work at hand introduces a technique for imposing Dirichlet boundary conditions directly without the need for indefinite systems. We use a formulation on the deformation gradient—also at small strains—and employ the Green’s operator associated to the vector Laplacian. Then, we develop the Moulinec–Suquet discretization for Dirichlet boundary conditions—requiring carefully selected weights at boundary points—and discuss the seamless integration into existing FFT-based computational homogenization codes based on dedicated discrete sine/cosine transforms. The article culminates with a series of well-chosen numerical examples demonstrating the capabilities of the introduced technology.

Stochastic electromechanical bidomain model *

We analyse a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs the dynamics of ionic quantities, intra and extra-cellular potentials, and linearised elasticity equations. We introduce a framework called the active strain decomposition, which factors the material gradient of deformation into an active (electrophysiology-dependent) part and an elastic (passive) part, to capture the coupling between muscle contraction, biochemical reactions, and electric activity. Under the assumption of linearised elastic behaviour and a truncation of the nonlinear diffusivities, we propose a stochastic electromechanical bidomain model, and establish the existence of weak solutions for this model. To prove existence through the convergence of approximate solutions, we employ a stochastic compactness method in tandem with an auxiliary non-degenerate system and the Faedo–Galerkin method. We utilise a stochastic adaptation of de Rham’s theorem to deduce the weak convergence of the pressure approximations.

Analytic rotation-invariant modelling of anisotropic finite elements

Anisotropic hyperelastic distortion energies are used to solve many problems in fields like computer graphics and engineering with applications in shape analysis, deformation, design, mesh parameterization, biomechanics and more. However, formulating a robust anisotropic energy that is low-order and yet sufficiently non-linear remains a challenging problem for achieving the convergence promised by Newton-type methods in numerical optimization. In this paper, we propose a novel analytic formulation of an anisotropic energy that is smooth everywhere, low-order, rotationally-invariant and at-least twice differentiable. At its core, our approach utilizes implicit rotation factorizations with invariants of the Cauchy-Green tensor that arises from the deformation gradient. The versatility and generality of our analysis is demonstrated through a variety of examples, where we also show that the constitutive law suggested by the anisotropic version of the well-known As-Rigid-As-Possible energy is the foundational parametric description of both passive and active elastic materials. The generality of our approach means that we can systematically derive the force and force-Jacobian expressions for use in implicit and quasistatic numerical optimization schemes, and we can also use our analysis to rewrite, simplify and speedup several existing anisotropic and isotropic distortion energies with guaranteed inversion-safety.