Continuum Mechanics Model Research Articles

Constitutive modeling of viscoplasticity including phase transformations for graded thermo‐mechanical processing

AbstractTailored materials are of particular interest in modern engineering applications due to their potentially high strength and residual ductility. During hot forming processes, such as press hardening, microstructure transformations in low carbon steels are considered to achieve the production of tailored components. The relevant microstructure mechanisms during thermo‐mechanical processing are visco‐plastic deformation and phase transformations. Thus, we present an extension to an existing thermodynamically consistent continuum mechanical model. The evolution of phase transformations under inhomogeneous temperature loading in a homogeneous tensile specimen are illustrated.

Physically intuitive continuum mechanics model for quartz crystal microbalance: Viscoelasticity of rubbery polymers at MHz frequencies

AbstractEmploying a quartz crystal microbalance (QCM) as a MHz‐viscoelastic sensor requires extracting information from higher harmonics beyond the Sauerbrey limit, which can be problematic for rubbery polymer films that are highly dissipative because of the onset of anharmonic side bands and film resonance. Data analysis for QCM can frequently obscure the underlying physics or involve approximations that tend to break down at higher harmonics. In this study, modern computational tools are leveraged to solve a continuum physics model for the QCM's acoustic shear wave propagation through a polymer film with zero approximations, retaining the physical intuition of how the experimental signal connects to the shear modulus of the material. The resulting set of three coupled equations are solved numerically to fit experimental data for the resonance frequency Δfn and dissipation ΔΓn shifts as a function of harmonic number n, over an extended harmonic range approaching film resonance. This allows the frequency‐dependent modulus of polymer films at MHz frequencies, modeled as linear on a log–log scale, to be determined for rubbery polybutadiene (PB) and polydimethylsiloxane (PDMS) films, showing excellent agreement with time–temperature shifted rheometry data from the literature.

A cell-centered implicit-explicit Lagrangian scheme for a unified model of nonlinear continuum mechanics on unstructured meshes

A cell-centered implicit-explicit updated Lagrangian finite volume scheme on unstructured grids is proposed for a unified first-order hyperbolic formulation of continuum fluid and solid mechanics, namely the Godunov-Peshkov-Romenski (GPR) model. The scheme provably respects the stiff relaxation limits of the continuous model at the fully discrete level, thus it is asymptotic preserving. Furthermore, the GCL is satisfied by a compatible discretization that makes use of a nodal solver to compute vertex-based fluxes that are used both for the motion of the computational mesh as well as for the time evolution of the governing PDEs. Second order of accuracy in space is achieved using a TVD piecewise linear reconstruction, while an implicit-explicit (IMEX) Runge-Kutta time discretization allows the scheme to obtain higher accuracy also in time. Particular care is devoted to the design of a stiff ODE solver, based on approximate analytical solutions of the governing equations, that plays a crucial role when the visco-plastic limit of the model is approached. We demonstrate the accuracy and robustness of the scheme on a wide spectrum of material responses covered by the unified continuum model that includes inviscid hydrodynamics, viscous heat conducting fluids, elastic and elasto-plastic solids in multidimensional settings.

Origami Spring-Inspired Shape Morphing for Flexible Robotics.

Flexible robotics are capable of achieving various functionalities by shape morphing, benefiting from their compliant bodies and reconfigurable structures. In this study, we construct and study a class of origami springs generalized from the known interleaved origami spring, as promising candidates for shape morphing in flexible robotics. These springs are found to exhibit nonlinear stretch-twist coupling and linear/nonlinear mechanical response in the compression/tension region, analyzed by the demonstrated continuum mechanics models, experiments, and finite element simulations. To improve the mechanical performance such as the damage resistance, we establish an origami rigidization method by adding additional creases to the spring system. Guided by the theoretical framework, we experimentally realize three types of flexible robotics-origami spring ejectors, crawlers, and transformers. These robots show the desired functionality and outstanding mechanical performance. The proposed concept of origami-aided design is expected to pave the way to facilitate the diverse shape morphing of flexible robotics.

Effective interface conditions for continuum mechanical models describing the invasion of multiple cell populations through thin membranes

We consider a continuum mechanical model for the migration of multiple cell populations through parts of tissue separated by thin membranes. In this model, cells belonging to different populations may be characterised by different proliferative abilities and mobility, which may vary from part to part of the tissue, as well as by different invasion potentials within the membranes. The original transmission problem, consisting of a set of mass balance equations for the volume fraction of cells of every population complemented with continuity of stresses and mass flux across the surfaces of the membranes, is then reduced to a limiting transmission problem whereby each thin membrane is replaced by an effective interface. In order to close the limiting problem, a set of biophysically-consistent transmission conditions is derived through a formal asymptotic method. Models based on such a limiting transmission problem may find fruitful application in a variety of research areas in the biological and medical sciences, including developmental biology, immunology and cancer growth and invasion.

On the Mathematical Modeling of Slender Biomedical Continuum Robots.

The passive, mechanical adaptation of slender, deformable robots to their environment, whether the robot be made of hard materials or soft ones, makes them desirable as tools for medical procedures. Their reduced physical compliance can provide a form of embodied intelligence that allows the natural dynamics of interaction between the robot and its environment to guide the evolution of the combined robot-environment system. To design these systems, the problems of analysis, design optimization, control, and motion planning remain of great importance because, in general, the advantages afforded by increased mechanical compliance must be balanced against penalties such as slower dynamics, increased difficulty in the design of control systems, and greater kinematic uncertainty. The models that form the basis of these problems should be reasonably accurate yet not prohibitively expensive to formulate and solve. In this article, the state-of-the-art modeling techniques for continuum robots are reviewed and cast in a common language. Classical theories of mechanics are used to outline formal guidelines for the selection of appropriate degrees of freedom in models of continuum robots, both in terms of number and of quality, for geometrically nonlinear models built from the general family of one-dimensional rod models of continuum mechanics. Consideration is also given to the variety of actuators found in existing designs, the types of interaction that occur between continuum robots and their biomedical environments, the imposition of constraints on degrees of freedom, and to the numerical solution of the family of models under study. Finally, some open problems of modeling are discussed and future challenges are identified.

Deep penetration of low velocity metal penetrators into ice

The paper summarizes the results of research of the destruction of an ice block with cylindrical and spherical penetrators at low velocity (≤325 m/s). The behaviour of ice at high strain rates is described by an elastic–plastic model of continuum mechanics. Numerical modelling of penetration is performed with IMPACT computer code. Algorithms of splitting nodes and destroying elements in a Lagrangian numerical method were modified to solve problems of penetration and perforation. The fracturing is described by a deterministic fracture model. Crack paths are examined; the damage is predicted and compared with existing experimental results. It was found that in the subsonic range of initial velocities the penetration time did not exceed 0.3 ms. Examination of the penetrators’ shapes showed that they were not plastic deformed.

Symmetric gradient Sobolev spaces endowed with rearrangement-invariant norms

A unified approach to embedding theorems for Sobolev type spaces of vector-valued functions, defined via their symmetric gradient, is proposed. The Sobolev spaces in question are built upon general rearrangement-invariant norms. Optimal target spaces in the relevant embeddings are determined within the class of all rearrangement-invariant spaces. In particular, all symmetric gradient Sobolev embeddings into rearrangement-invariant target spaces are shown to be equivalent to the corresponding embeddings for the full gradient built upon the same spaces. A sharp condition for embeddings into spaces of uniformly continuous functions, and their optimal targets, are also exhibited. By contrast, these embeddings may be weaker than the corresponding ones for the full gradient. Related results, of independent interest in the theory of symmetric gradient Sobolev spaces, are established. They include global approximation and extension theorems under minimal assumptions on the domain. A formula for the K-functional, which is pivotal for our method based on reduction to one-dimensional inequalities, is provided as well. The case of symmetric gradient Orlicz-Sobolev spaces, of use in mathematical models in continuum mechanics driven by nonlinearities of non-power type, is especially focused.

A Physiology-Guided Classification of Active-Stress and Active-Strain Approaches for Continuum-Mechanical Modeling of Skeletal Muscle Tissue.

The well-established sliding filament and cross-bridge theory explain the major biophysical mechanism responsible for a skeletal muscle's active behavior on a cellular level. However, the biomechanical function of skeletal muscles on the tissue scale, which is caused by the complex interplay of muscle fibers and extracellular connective tissue, is much less understood. Mathematical models provide one possibility to investigate physiological hypotheses. Continuum-mechanical models have hereby proven themselves to be very suitable to study the biomechanical behavior of whole muscles or entire limbs. Existing continuum-mechanical skeletal muscle models use either an active-stress or an active-strain approach to phenomenologically describe the mechanical behavior of active contractions. While any macroscopic constitutive model can be judged by it's ability to accurately replicate experimental data, the evaluation of muscle-specific material descriptions is difficult as suitable data is, unfortunately, currently not available. Thus, the discussions become more philosophical rather than following rigid methodological criteria. Within this work, we provide a extensive discussion on the underlying modeling assumptions of both the active-stress and the active-strain approach in the context of existing hypotheses of skeletal muscle physiology. We conclude that the active-stress approach resolves an idealized tissue transmitting active stresses through an independent pathway. In contrast, the active-strain approach reflects an idealized tissue employing an indirect, coupled pathway for active stress transmission. Finally the physiological hypothesis that skeletal muscles exhibit redundant pathways of intramuscular stress transmission represents the basis for considering a mixed-active-stress-active-strain constitutive framework.

Design of active materials distributions for four-dimensional printing based on multi-material topology optimization

Printed active composites (PACs) are capable of deforming from an initial shape to a target shape via spatial arrangements of active materials within a passive matrix. Multi-material polymer printers allow precise placement of multiple materials in the design space. However, single active material is difficult to satisfy the demand in the high-precision matching of more complex target shape. Hence, a multi-material topology optimization approach for the design of PACs is proposed to achieve a target shape under a given stimulus. A multi-material interpolation function for active materials is established, and continuum mechanics modeling is used for simulating active material behaviors on a voxel basis to compute deformations for a given material distribution. The adjoint method is used for sensitivity analysis. Numerical simulations, of different target shapes with the same initial shape, verify the effectiveness of the method. The proposed method is feasible in the conceptual designs of PAC material distributions benefit with stable convergence, ease of implementation, and low computational costs.

A finite element method for modeling surface growth and resorption of deformable solids

This work explores a continuum-mechanical model for a body simultaneously undergoing finite deformation and surface growth/resorption. This is accomplished by defining the kinematics as well as the set of material points that constitute the domain of a physical body at a given time in terms of an evolving reference configuration. The implications of spatial and temporal discretization are discussed, and an extension of the Arbitrary Lagrangian–Eulerian finite element method is proposed to enforce the resulting balance laws on the grown/resorbed body in two spatial dimensions. Representative numerical examples are presented to highlight the predictive capabilities of the model and the numerical properties of the proposed solution method.

A quasi-nonlocal coupling method for bond-based peridynamics with classical continuum mechanics

PurposeThe purpose of this paper is to propose a novel quasi-nonlocal coupling of the bond-based peridynamic model with the classical continuum mechanics model to fully take advantage of their merits and be free of ghost forces.Design/methodology/approachThis study reconstructs a total energy functional by introducing a coupling parameter that alters only the nonlocal interactions in the coupling region rather than the whole region and a modified elasticity tensor that affects the local interactions. Then, the consistency of force patch test is enforced in the coupling region to completely eliminate the ghost force in a general energy-based coupling scheme. For a one-dimensional problem, these coupling parameters are further determined through an energy patch test to preserve the energy equivalence or through an l1-regularization. And, for a two- or three-dimensional problem, depending on the existence of a solution to the discretized force patch test, they are determined through an l1-minimization or l1-regularization.FindingsOne- and two-dimensional numerical examples under affine deformation have been conducted to verify the accuracy of the quasi-nonlocal coupling method, which exhibits no ghost force. Moreover, the coupling model can reproduce almost the same deformation behaviors of points near the crack for a cracked plate under tension as that from a pure peridynamic model, the former with a rather low computational cost and an easier application of boundary conditions.Originality/valueThis work is aiming at getting over long-standing ghost force issues in the energy-based coupling scheme. The numerical results from the cracked plate problem are exhibited promising extension to dynamic problems.

On a phase-field approach to model fracture of small intestine walls

We address anisotropic elasticity and fracture in small intestine walls (SIWs) with both experimental and computational methods. Uniaxial tension experiments are performed on porcine SIW samples with varying alignments and quantify their nonlinear elastic anisotropic behavior. Fracture experiments on notched SIW strips reveal a high sensitivity of the crack propagation direction and the failure stress on the tissue orientation. From a modeling point of view, the observed anisotropic elastic response is studied with a continuum mechanical model stemming from a strain energy density with a neo-Hookean component and an anisotropic component with four families of fibers. Fracture is addressed with the phase-field approach, featuring two-fold anisotropy in the fracture toughness. Elastic and fracture model parameters are calibrated based on the experimental data, using the maximum and minimum limits of the experimental stress-stretch data set. A very good agreement between experimental data and computational results is obtained, the role of anisotropy being effectively captured by the proposed model in both the elastic and the fracture behavior. Statement of significanceThis article reports a comprehensive experimental data set on the mechanical failure behavior of small intestinal tissue, and presents the corresponding protocols for preparing and testing the samples. On the one hand, the results of this study contribute to the understanding of small intestine mechanics and thus to understanding of load transfer mechanisms inside the tissue. On the other hand, these results are used as input for a phase-field modelling approach, presented in this article. The presented model can reproduce the mechanical failure behavior of the small intestine in an excellent way and is thus a promising tool for the future mechanical description of diseased small intestinal tissue.

Peridynamic bond‐associated correspondence model: Wave dispersion property

AbstractThe peridynamic bond‐associated material correspondence model is a reformulation of the conventional correspondence model, in which the concept of bond‐associated states were introduced to inherently remove the material instability. Numerical studies, in terms of deformation analysis, have shown the effectiveness of the bond‐associated model in removing the material instability. In this study, the bond‐associated model is further examined concerning its wave dispersion property. Detailed procedure to derive the wave dispersion relation in this model is outlined. Different combinations of the peridynamic material point horizon and the bond‐associated horizon are studied for all three geometric dimensions. Potential effects of weight function on the wave dispersion property are also investigated. It is found that the bond‐associated material correspondence model can effectively remove the material instability for practical applications. However, for all one‐dimensional cases studied, except the conventional model, the initial slopes of all the wave dispersion curves are slightly larger than that predicted by the classical continuum mechanics model. For two‐dimensional analysis, material instability still exists for the combination of the peridynamic material point horizon being twice of the mesh spacing and the bond‐associated horizon being three times of mesh spacing. But compared to the conventional model, the material instability for this case was slightly improved as the intersections of the wave dispersion curve with the wavenumber axis were reduced from two to one. For the three representative weight functions studied, it is found that the weight function does not significantly change the wave dispersion relation in the bond‐associated material correspondence model, except for the one‐dimensional cases.

A machine learning approach to estimate the strain energy absorption in expanded polystyrene foams

Traditional modeling of mechanical energy absorption due to compressive loadings in expanded polystyrene foams involves mathematical descriptions that are derived from stress/strain continuum mechanics models. Nevertheless, most of those models are either constrained using the strain as the only variable to work at large deformation regimes and usually neglect important parameters for energy absorption properties such as the material density or the rate of the applying load. This work presents a neural-network-based approach that produces models that are capable to map the compressive stress response and energy absorption parameters of an expanded polystyrene foam by considering its deformation, compressive loading rates, and different densities. The models are trained with ground-truth data obtained in compressive tests. Two methods to select neural network architectures are also presented, one of which is based on a Design of Experiments strategy. The results show that it is possible to obtain a single artificial neural networks model that can abstract stress and energy absorption solution spaces for the conditions studied in the material. Additionally, such a model is compared with a phenomenological model, and the results show than the neural network model outperforms it in terms of prediction capabilities, since errors around 2% of experimental data were obtained. In this sense, it is demonstrated that by following the presented approach is possible to obtain a model capable to reproduce compressive polystyrene foam stress/strain data, and consequently, to simulate its energy absorption parameters.

Identification of mechanical properties of arteries with certification of global optimality

In this study, we consider identification of parameters in a non-linear continuum-mechanical model of arteries by fitting the models response to clinical data. The fitting of the model is formulated as a constrained non-linear, non-convex least-squares minimization problem. The model parameters are directly related to the underlying physiology of arteries, and correctly identified they can be of great clinical value. The non-convexity of the minimization problem implies that incorrect parameter values, corresponding to local minima or stationary points may be found, however. Therefore, we investigate the feasibility of using a branch-and-bound algorithm to identify the parameters to global optimality. The algorithm is tested on three clinical data sets, in each case using four increasingly larger regions around a candidate global solution in the parameter space. In all cases, the candidate global solution is found already in the initialization phase when solving the original non-convex minimization problem from multiple starting points, and the remaining time is spent on increasing the lower bound on the optimal value. Although the branch-and-bound algorithm is parallelized, the overall procedure is in general very time-consuming.

In vivo parameter identification in arteries considering multiple levels of smooth muscle activity

In this paper an existing in vivo parameter identification method for arteries is extended to account for smooth muscle activity. Within this method a continuum-mechanical model, whose parameters relate to the mechanical properties of the artery, is fit to clinical data by solving a minimization problem. Including smooth muscle activity in the model increases the number of parameters. This may lead to overparameterization, implying that several parameter combinations solve the minimization problem equally well and it is therefore not possible to determine which set of parameters represents the mechanical properties of the artery best. To prevent overparameterization the model is fit to clinical data measured at different levels of smooth muscle activity. Three conditions are considered for the human abdominal aorta: basal during rest; constricted, induced by lower-body negative pressure; and dilated, induced by physical exercise. By fitting the model to these three arterial conditions simultaneously a unique set of model parameters is identified and the model prediction agrees well with the clinical data.

Ultrafast strain propagation and acoustic resonances in nanoscale bilayer systems.

Ultrafast structural probing has greatly enhanced our understanding of the coupling of atomic motion to electronic and phononic degrees-of-freedom in quasi-bulk materials. In bi- and multilayer model systems, additionally, spatially inhomogeneous relaxation channels are accessible, often governed by pronounced interfacial couplings and local excitations in confined geometries. Here, we systematically explore the key dependencies of the low-frequency acoustic phonon spectrum in an elastically mismatched metal/semiconductor bilayer system optically excited by femtosecond laser pulses. We track the spatiotemporal strain wave propagation in the heterostructure employing a discrete numerical linear chain simulation and access acoustic wave reflections and interfacial couplings with a phonon mode description based on a continuum mechanics model. Due to the interplay of elastic properties and mass densities of the two materials, acoustic resonance frequencies of the heterostructure significantly differ from breathing modes in monolayer films. For large acoustic mismatch, the spatial localization of phonon eigenmodes is derived from analytical approximations and can be interpreted as harmonic oscillations in decoupled mechanical resonators.

Mechanical models and numerical simulations in nanomechanics: A review across the scales

This work gives an overview of the theoretical background and of the numerical modelling framework used to describe the mechanical properties and the response of materials on scales ranging from the atomistic, through the microstructure and all the way up to the macroscale. In order to describe the dual nature of the structure of matter, which is continuous when viewed at large length scales and discrete when viewed at the atomic scale, plus the interdependence of these scales, multiscale modelling is required to complement the continuum and the atomistic models. More specifically, what we aim for in this review is to present and discuss the following basic conceptual models, as well as the methodologies that accompany them: (a) discrete models such as ab initio, atomistic / molecular, mesoscopic; (b) continuum mechanics models (CMM) comprising pure CMM, non-local elasticity CMM, higher-order strain gradient and higher-order nonlocal strain gradient elasticity CMM, and surface elasticity CMM; (c) multiscale material models (MMM). Since the field of nanomechanics is currently a rapidly expanding research area, the presented state-of-the art is by no means exhaustive. It simply outlines the research efforts that go behind formulating numerical models for the solution of problems in nanomechanics. Despite the advantages that boundary element methods (BEM) have in solving problems at the physical scale, either as stand-alone or in combination with finite element methods (FEM), their application to multiscale modelling is still limited, despite the promise they seem to hold.

Peridynamic modeling with energy-based surface correction for fracture simulation of random porous materials

This study presents a peridynamic model with an improved energy-based surface correction to simulate fractures of random porous materials. There exists a surface effect due to incomplete neighborhoods of boundary or near-boundary points in peridynamics. A surface correction approach is proposed to correct the surface effect on or near the boundaries, including outer boundaries of the materials and boundaries of inner pores. The correction factor is obtained based on the equivalence between the strain energy density in classical continuum mechanics and the elastic energy density in peridynamics. The algorithm procedure for the corrected model based on the finite element analysis is brought forward in detail. The availability of the proposed corrected peridynamic model is verified by comparing the results against the results by the classical continuum mechanics model. Moreover, the effect of the surface correction on the crack growth path and the impact of random distributions of pores on the fractures of porous materials are also investigated. Numerical results demonstrate that the present method can effectively correct the surface effect of random porous materials, especially around the boundary of inner pores, where shows a more obvious surface effect.