Continuum Bodies Research Articles

A single-surface multi-failure strength domain for masonry

In this paper, we propose a simple approach to derive a single-surface multi-failure strength domain for the in-plane behaviour of masonry. The approach, that lays the basis for homogeneous continuum model developments relies on micro-mechanical analyses employing a block-based model for masonry in which blocks, modelled as finite strength continuum bodies, interact through zero-thickness interfaces. In order to derive the strength domain, firstly three failure mechanism typologies are identified, namely crushing failure, joint failure and mixed joint-block failure. Then, the limit surface for each mechanism is obtained relying on limit equilibrium considerations, also introducing a novel rational treatment of the mixed mechanism. Accordingly, a multi-surface strength domain is built by intersecting all the limit surfaces. Finally, such multi-surface strength domain is reduced to a single-surface one exploiting the RealSoftMax function, which allows to preserve the multi-failure nature of the approach, i.e. the explicit distinction between all the failure mechanisms. Following the proposed procedure, the resulting strength domain inherits the material parameters characterizing the block-based model. A finite element block-based model and available experimental data are employed to validate the proposed strength domain. The good agreement obtained with reference results confirms the soundness of the approach.

Performance enhancement of the soft robotic segment for a trunk-like arm.

Trunk-like continuum robots have wide applications in manipulation and locomotion. In particular, trunk-like soft arms exhibit high dexterity and adaptability very similar to the creatures of the natural world. However, owing to the continuum and soft bodies, their performance in payload and spatial movements is limited. In this paper, we investigate the influence of key design parameters on robotic performance. It is verified that a larger workspace, lateral stiffness, payload, and bending moment could be achieved with adjustments to soft materials' hardness, the height of module segments, and arrayed radius of actuators. Especially, a 55% increase in arrayed radius would enhance the lateral stiffness by 25% and a bending moment by 55%. An 80% increase in segment height would enlarge 112% of the elongation range and 70 % of the bending range. Around 200% and 150% increments in the segment's lateral stiffness and payload forces, respectively, could be obtained by tuning the hardness of soft materials. These relations enable the design customization of trunk-like soft arms, in which this tapering structure ensures stability via the stocky base for an impact reduction of 50% compared to that of the tip and ensures dexterity of the long tip for a relatively larger bending range of over 400% compared to that of the base. The complete methodology of the design concept, analytical models, simulation, and experiments is developed to offer comprehensive guidelines for trunk-like soft robotic design and enable high performance in robotic manipulation.

Distance Minimizing-Based Data-Driven Computational Plasticity Method with Fixed Dataset

A data-driven computational plasticity method based on the distance minimizing framework is proposed in this paper. In this method, the internal variables in conventional plasticity are abandoned and a fixed dataset considering path-dependent behaviors of materials is constructed. With the fixed dataset, a stress correspondence method is developed to compute the plastic strain of every integration point at each load step, and a data-driven classification model for yielding is constructed to rapidly determine the yield status of each point in the method. Moreover, a symmetric mapping method is developed to accurately determine the stress–strain state of the integration point under unloading or inverse loading conditions. Several representative examples are presented to show the capability of the proposed method. Numerical results of two- and three-dimensional truss structures and three-dimensional continuum bodies demonstrate the high efficiency and accuracy of the proposed data-driven computational plasticity method.

Size dependent generalized thermoelasticity: Green-Lindsay theory with modified strain gradient theory

The present work is a development of the Green-Lindsay (GL) thermoelasticity theory for micro continuum bodies. The micro continuum modified strain gradient theory is used here based on the Green-Lindsay theory of the micro continuum media. In fact, the present work is a combination of Green-Lindsay thermoelasticity theory with modified strain gradient continuum theory. All assumptions of GL and modified strain gradient theory are satisfied and the final general forms of constitutive relations and also heat conduction equation are extracted. As an application of the derived formulation, a one-dimensional microlayer is considered and the newly derived equations are applied to it. Finally, the governing equations of coupled generalized thermoelasticity of the microlayer are solved using the Chebyshev collocation method. Effects of microscale parameters appearing in modified strain gradient theory on the dynamic thermo-elastic response of microlayer are investigated.

Geotechnical particle finite element method for modeling of soil-structure interaction under large deformation conditions

The possibilities of the particle finite element method (PFEM) for modeling geotechnical problems are increasingly evident. PFEM is a numerical approach to solve large displacement and large strain continuum problems that are beyond the capabilities of classical finite element method (FEM). In PFEM, the computational domain is reconfigured for optimal solution by frequent remeshing and boundary updating. PFEM inherits many concepts, such as a Lagrangian description of continuum, from classic geomechanical FEM. This familiarity with more popular numerical methods facilitates learning and application. This work focuses on G-PFEM, a code specifically developed for the use of PFEM in geotechnical problems. The article has two purposes. The first is to give the reader an overview of the capabilities and main features of the current version of the G-PFEM and the second is to illustrate some of the newer developments of the code. G-PFEM can solve coupled hydro-mechanical static and dynamic problems involving the interaction of solid and/or deformable bodies. Realistic constitutive models for geomaterials are available, including features, such as structure and destructuration, which result in brittle response. The solutions are robust, solidly underpinned by numerical technology including mixed-field formulations, robust and mesh-independent integration of elastoplastic constitutive models and a rigorous and flexible treatment of contact interactions. The novel features presented in this work include the contact domain technique, a natural way to capture contact interactions and impose contact constraints between different continuum bodies, as well as a new simplified formulation for dynamic impact problems. The code performance is showcased by the simulation of several soil-structure interaction problems selected to highlight the novel code features: a rigid footing insertion in soft rock, pipeline insertion and subsequent lateral displacement on over-consolidated clay, screw-pile pull-out and the dynamic impact of a free-falling spherical penetrometer into clay.

FrictionalMonolith

We propose FrictionalMonolith , a monolithic pressure-friction-contact solver for more accurately, robustly, and efficiently simulating two-way interactions of rigid bodies with continuum granular materials or inviscid liquids. By carefully formulating the components of such systems within a single unified minimization problem, our solver can simultaneously handle unilateral incompressibility and implicit integration of friction for the interior of the continuum, frictional contact resolution among the rigid bodies, and mutual force exchanges between the continuum and rigid bodies. Our monolithic approach eliminates various problematic artifacts in existing weakly coupled approaches, including loss of volume in the continuum material, artificial drift and slip of the continuum at solid boundaries, interpenetrations of rigid bodies, and simulation instabilities. To efficiently handle this challenging monolithic minimization problem, we present a customized solver for the resulting quadratically constrained quadratic program that combines elements of staggered projections, augmented Lagrangian methods, inexact projected Newton, and active-set methods. We demonstrate the critical importance of a unified treatment and the effectiveness of our proposed solver in a range of practical scenarios.

Skeletonizing the Dynamics of Soft Continuum Body from Video.

Soft continuum bodies have demonstrated their effectiveness in generating flexible and adaptive functionalities by capitalizing on the rich deformability of soft material. Compared with a rigid-body robot, it is in general difficult to model and emulate the morphology dynamics of a soft continuum body. In addition, a soft continuum body potentially has an infinite degree of freedom, requiring considerable labor to manually annotate its dynamics from external sensory data such as video. In this study, we propose a novel noninvasive framework for automatically extracting the skeletal dynamics from video of a soft continuum body and show the applications and effectiveness of our framework. First, we demonstrate that our framework can extract skeletal dynamics from animal videos, which can be effectively utilized for the analysis of soft continuum body including animal motion. Next, we focus on a soft continuum arm, a commonly used platform in soft robotics, and evaluate the potential information-processing capability. Normally, to control such a high-dimensional system, it is necessary to introduce many sensors to completely capture the motion dynamics, causing the deterioration of the material's softness. We illustrate that the evaluation of the memory capacity and sensory reconstruction error enables us to verify the minimum number of sensors sufficient for fully grasping the state dynamics, which is highly useful in designing a sensor arrangement for a soft robot. Also, we release the software developed in this study as open source for biology and soft robotics communities, which contributes to automating the annotation process required for the motion analysis of soft continuum bodies.

A Mesh-Free Approach for Multiscale Modeling in Continuum–Granular Systems

Geotechnical systems often examine interactions that occur between continuum bodies and granular soils. The systems and interactions can be accurately simulated by using multiscale coupling approaches. The model for the continuum bodies is often constructed into a mesh. The meshing, however, is time consuming for a huge spatial system and if distorted is subject to adjustments. A mesh-free approach can be used to eliminate these drawbacks. In this study, a mesh-free approach for simulating continuum–granular systems is presented. This approach combines element-free Galerkin (EFG) and discrete element (DE) methods to approximate the interactions. The capabilities of the coupled EFG–DE method are validated through its solving two example problems: the cantilever beam–disc system and Cundall’s Nine Disc Test. The proposed approach appears to be an efficient and promising tool to model multiscale, multibody contacting problems.

Model-Free Soft-Structure Reconstruction for Proprioception Using Tactile Arrays

Continuum body structures provide unique opportunities for soft robotics, with the infinite degrees of freedom enabling unconstrained and highly adaptive exploration and manipulation. However, the infinite degrees of freedom of continuum bodies makes sensing (both intrinsically and extrinsically) challenging. To address this, in this letter, we propose a model-free method for sensorizing tentacle-like continuum soft-structures using an array of spatially arranged capacitive tactile sensors. By using visual tracking, the relationship between the tactile response and the three-dimensional shape of the continuum soft-structure can be learned. A dataset of 15 000 random soft-body postures was used, with recorded camera-tracked positions logged synchronously to the tactile sensor responses. This was used to train a neural network that can predict posture. We show that it is possible to achieve proprioceptive awareness over all three axis of motion in space, reconstructing the body structure and inferring the soft body head's pose with an average accuracy of ≈1 mm in comparison to the visual tracked counterpart. To demonstrate the capabilities of the system, we perform random exploration of environments limiting the work-space of the sensorized robot. We find the method capable to autonomously reconstruct the reachable morphology of the environment without the need of external sensing units.

On stress/strain state in a rotating disk

In the framework of mechanics of continuum bodies, the problem of stress/strain state in a high-speed rotating disk of constant thickness has been considered. The material of the disk is assumed to be homogeneous, elastic/perfectly-plastic. In the plastic zone, the stresses and plastic strains are related by some associated law similar to the one employed in deformation theory of plasticity. The general algorithm of the solution covers any smooth plasticity function. At some steps of the algorithm, it is possible to get analytical expressions, particularly, for the quadratic Mises yield criterion. For the given model, the notion of control parameters (external and internal) has been introduced. The allowable boundaries of external parameters have been defined as well. For some states of the disk, the coherent values of external parameters have been obtained. The results are represented graphically to show various states of the disk. The usage of piecewise plasticity functions has been briefly discussed. The results obtained can be used in preliminary engineering design and related numerical codes.

The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies.

The paper discusses the problem of manufacturability of the minimum compliance designs of the structural elements made of two kinds of inhomogeneous materials: the isotropic and cubic. In both the cases the unit cost of the design is assumed as equal to the trace of the Hooke tensor. The Isotropic Material Design (IMD) delivers the optimal distribution of the bulk and shear moduli within the design domain. The Cubic Material Design (CMD) leads to the optimal material orientation and optimal distribution of the invariant moduli in the body made of the material of cubic symmetry. The present paper proves that the varying underlying microstructures (i.e., the representative volume elements (RVE) constructed of one or two isotropic materials) corresponding to the optimal designs constructed by IMD and CMD methods can be recovered by matching the values of the optimal moduli with the values of the effective moduli of the RVE computed by the theory of homogenization. The CMD method leads to a larger set of results, i.e., the set of pairs of optimal moduli. Moreover, special attention is focused on proper recovery of the microstructures in the auxetic sub-domains of the optimal designs.

Numerical homogenization of the Eshelby tensor at small strains

Numerical homogenization methods, such as the FE2 approach, are widely used to compute the effective physical properties of microstructured materials. Thereby, the macroscopic material law is replaced by the solution of a microscopic boundary value problem on a representative volume element in conjunction with appropriate averaging techniques. This concept can be extended to configurational or material quantities, like the Eshelby stress tensor, which are associated with configurational changes of continuum bodies. In this work, the focus is on the computation of the macroscopic Eshelby stress tensor within a small-strain setting. The macroscopic Eshelby stress tensor is defined as the volume average of its microscopic counterpart. On the microscale, the Eshelby stress tensor can be computed from quantities known from the solution of the physical microscopic boundary value problem. However, in contrast to the physical quantities of interest, i.e. stress and strain, the Eshelby stress tensor is sensitive to rigid body rotations of the representative volume element. In this work, it is demonstrated how this must be taken into account in the computation of the macroscopic Eshelby stress tensor. The theoretical findings are illustrated by a benchmark simulation and further simulation results indicate the microstructural influence on the macroscopic configurational forces.

Continuum and discrete models for unbalanced woven fabrics

The classical models used for describing the mechanical behavior of woven fabrics do not fully account for the whole set of phenomena that occur during the testing of such materials. This lack of precision is mainly due to the absence of energy terms related to the microstructural properties of the fabric and, in particular, to the bending stiffness of the yarns. The importance of the bending stiffness on the overall mechanical behavior of woven reinforcements, if already essential for the complete description of balanced fabrics, becomes even more important in the case of unbalanced ones. In this paper it is shown that the unbalance in the bending stiffnesses of the warp and weft yarns produces macroscopic effects that are extremely visible: we mention, for example, the asymmetric S-shape of a woven interlock subjected to a Bias Extension Test. We propose to introduce a constrained micromorphic model and, simultaneously, a discrete model that are both able to account for (i) the angle variation between warp and weft tows, (ii) the unbalance in the bending stiffness of the yarns and (iii) the relative slipping of the tows. The introduced constrained micromorphic model is rigorously framed in the spirit of the Principle of Virtual Work for the study of the equilibrium of continuum bodies. A suitable constraint is introduced in such micromorphic model by means of Lagrange multipliers in the strain energy density and the resulting constrained model is seen to be a particular second gradient one. The main advantage of using such constrained micromorphic model is that the kinematical and traction boundary conditions that can be imposed on some sub-portions of the boundary of the considered body take a natural and unique meaning. The discrete model is set up by opportunely interconnecting Euler–Bernoulli beams with different bending stiffnesses in the two directions by means of rotational and translational elastic springs. The main strength of such discrete model is that the slipping of the tows is described in a rather realistic way. Suitable numerical simulations are presented for both the continuum and the discrete models and a comparison between the simulations and the experimental results is performed showing a definitely good agreement.

Rational ANCF Thin Plate Finite Element

In this paper, a rational absolute nodal coordinate formulation (RANCF) thin plate element is developed and its use in the analysis of curved geometry is demonstrated. RANCF finite elements are the rational counterpart of the nonrational absolute nodal coordinate formulation (ANCF) finite elements which employ rational polynomials as basis or blending functions. RANCF finite elements can be used in the accurate geometric modeling and analysis of flexible continuum bodies with complex geometrical shapes that cannot be correctly described using nonrational finite elements. In this investigation, the weights, which enter into the formulation of the RANCF finite element and form an additional set of geometric parameters, are assumed to be nonzero constants in order to accurately represent the initial geometry and at the same time preserve the desirable ANCF features, including a constant mass matrix and zero centrifugal and Coriolis generalized inertia forces. A procedure for defining the control points and weights of a Bezier surface defined in a parametric form is used in order to be able to efficiently create RANCF/ANCF FE meshes in a straightforward manner. This procedure leads to a set of linear algebraic equations whose solution defines the RANCF coordinates and weights without the need for an iterative procedure. In order to be able to correctly describe the ANCF and RANCF gradient deficient FE geometry, a square matrix of position vector gradients is formulated and used to calculate the FE elastic forces. As discussed in this paper, the proposed finite element allows for describing exactly circular and conic sections and can be effectively used in the geometry and analysis modeling of multibody system (MBS) components including tires. The proposed RANCF finite element is compared with other nonrational ANCF plate elements. Several numerical examples are presented in order to demonstrate the use of the proposed RANCF thin plate element. In particular, the FE models of a set of rational surfaces, which include conic sections and tires, are developed.

Theoretical and numerical predictions of stretch-bending deformation behavior in composite sheet

Stretch-bending of fiber reinforced composite sheets is one of the thermo-mechanical forming techniques in composite materials manufacturing. Due to an anisotropic nature of composite materials, predicting their deformation behavior during combined stretch-bending is difficult. To analyze this behavior, a mathematical model for semi-cylinder forming of continuous and unidirectional fiber reinforced composite sheet is developed. Plies in the sheet are treated as continuum bodies that interact through weak viscous interfaces. The model depicts both spatial and temporal variations of stresses and strains in composite sheet. Based on this model, numerical examples on semi-cylinder forming of cross-ply and angle-ply symmetric composite sheets are given and the results show good agreement with the simulation results obtained by ABAQUS commercial software.

A constrained generalised- $$\alpha $$ α method for coupling rigid parallel chain kinematics and elastic bodies

A problem arises from combining flexible rotorcraft blades with stiffer mechanical links, which form a parallel kinematic chain. This paper introduces a method for solving index-3 differential algebraic equations for coupled stiff and elastic body systems with closed-loop kinematics. Rigid body dynamics and elastic body mechanics are independently described according to convenient mathematical measures. Holonomic constraint equations couple both the parallel chain kinematics and describe the coupling between the rigid and continuum bodies. Lagrange multipliers enforce the kinetic conditions for both sets of constraints. Additionally, to prevent numerical inaccuracy from inverting stiff mechanical matrices, a scaling factor normalises the dynamic tangential stiffness matrix. Finally, example tests show the verification of the algorithm with respect to existing computational tests and the accuracy of the model for cases relevant to the problem definition.

Kinematic modelling and configuration estimation for elastic wire using minimum sensors

Configuration estimation of elastic wire has been confined to small-sized continuum bodies. This work aims at configuration estimation algorithm of general elastic wires using minimum number of information (i.e. two magnetic sensor and length of wire) with consideration of gravity load of the elastic wire. The elastic wire is modelled as a kinematically redundant manipulator with multiple links and joints. An optimization algorithm minimizing the potential energy of the elastic wire is employed. The performance of the configuration estimation algorithm is proven by comparison of simulation and experiment results.

Energy-consistent time integration for nonlinear viscoelasticity

This paper is concerned with the numerical solution of the evolution equations of thermomechanical systems, in such a way that the scheme itself satisfies the laws of thermodynamics. Within this framework, we present a novel integration scheme for the dynamics of viscoelastic continuum bodies in isothermal conditions. This method intrinsically satisfies the laws of thermodynamics arising from the continuum, as well as the possible additional symmetries. The resulting solutions are physically accurate since they preserve the fundamental physical properties of the model. Furthermore, the method gives an excellent performance with respect to robustness and stability. Proof for these claims as well as numerical examples that illustrate the performance of the novel scheme are provided.

A torsion-free non-linear beam model

The governing equations of a novel structural model are derived from general balance equations and Lagrangian mechanics. The model represents beam-like slender bodies whose sections can withstand traction and shear forces, plus bending, but no torsional, moments. Although continuum bodies with such a behavior do not exist, multibody systems whose overall response can be described in such a way can be found. In fact, the motivation for this work is the study of certain types of polymers, which can be efficiently modeled as torsion-free rods instead of using long spring–mass chains, for which the proposed continuous model can serve as a basis for efficient finite element discretizations.

Variational Integrators and Energy-Momentum Schemes for Flexible Multibody Dynamics

This work contains a comparison between variational integrators and energy-momentum schemes for flexible multibody dynamics. In this connection, a specific “rotationless” formulation of flexible multibody dynamics is employed. Flexible components such as continuum bodies and geometrically exact beams and shells are discretized in space by using nonlinear finite element methods. The motion of the resulting discrete systems are governed by a uniform set of differential-algebraic equations (DAEs). This makes possible the application and comparison of previously developed structure-preserving methods for the numerical integration of the DAEs. In particular, we apply a specific variational integrator and an energy-momentum scheme. The performance of both integrators is assessed in the context of three representative numerical examples.