Showing papers on "Continuum mechanics published in 2022"

A mechanics‐informed artificial neural network approach in data‐driven constitutive modeling

Abstract: A mechanics‐informed artificial neural network approach for learning constitutive laws governing complex, nonlinear, elastic materials from strain–stress data is proposed. The approach features a robust and accurate method for training a regression‐based model capable of capturing highly nonlinear strain–stress mappings, while preserving some fundamental principles of solid mechanics. In this sense, it is a structure‐preserving approach for constructing a data‐driven model featuring both the form‐agnostic advantage of purely phenomenological data‐driven regressions and the physical soundness of mechanistic models. The proposed methodology enforces desirable mathematical properties on the network architecture to guarantee the satisfaction of physical constraints such as objectivity, consistency (preservation of rigid body modes), dynamic stability, and material stability, which are important for successfully exploiting the resulting model in numerical simulations. Indeed, embedding such notions in a learning approach reduces a model's sensitivity to noise and promotes its robustness to inputs outside the training domain. The merits of the proposed learning approach are highlighted using several finite element analysis examples. Its potential for ensuring the computational tractability of multi‐scale applications is demonstrated with the acceleration of the nonlinear, dynamic, multi‐scale, fluid‐structure simulation of the supersonic inflation dynamics of a parachute system with a canopy made of a woven fabric.

A review of size-dependent continuum mechanics models for micro- and nano-structures

Abstract: Recently, the mechanical behavior of micro-/nano-structures has sparked an ongoing debate, which leads to a fundamental question: what steps can be taken to investigate the mechanical characteristics of these structures, and characterize their performance? From the standpoint of the non-classical behavior of materials, size-dependent theories of micro-/nano-structures can be considered to analyze their mechanical behavior. The application of classical theories in the investigation of small-scale structures can lead to inaccurate results. Many studies have been published in the past few years, in which continuum mechanics models have been used to investigate micro-/nano-structures with different geometry such as rods, tubes, beams, plates, and shells. The mechanical behavior of these systems under different loading – resulting in vibration, wave propagation, bending, and buckling phenomena – is the focus of the review covered in this work. The present objective is to provide a detailed survey of the most significant literature on continuum mechanics models of micro-/nano-structures, and thus orient researchers in their future studies in this field of research.

A review of size-dependent continuum mechanics models for micro- and nano-structures

Abstract: Recently, the mechanical behavior of micro-/nano-structures has sparked an ongoing debate, which leads to a fundamental question: what steps can be taken to investigate the mechanical characteristics of these structures, and characterize their performance? From the standpoint of the non-classical behavior of materials, size-dependent theories of micro-/nano-structures can be considered to analyze their mechanical behavior. The application of classical theories in the investigation of small-scale structures can lead to inaccurate results. Many studies have been published in the past few years, in which continuum mechanics models have been used to investigate micro-/nano-structures with different geometry such as rods, tubes, beams, plates, and shells. The mechanical behavior of these systems under different loading – resulting in vibration, wave propagation, bending, and buckling phenomena – is the focus of the review covered in this work. The present objective is to provide a detailed survey of the most significant literature on continuum mechanics models of micro-/nano-structures, and thus orient researchers in their future studies in this field of research.

Finite element method for stress-driven nonlocal beams

Abstract: The bending behaviour of systems of straight elastic beams at different scales is investigated by the well-posed stress-driven nonlocal continuum mechanics. An effective computational methodology, based on nonlocal two-noded finite elements, is developed in order to take accurately into account long-range interactions present in the whole structural domain. The idea consists in partitioning the beam in subdomains and in observing that the nonlocal stress-driven convolution integral, equipped with the Helmholtz averaging kernel, can be equivalently formulated by expressing nonlocal bending interaction fields in terms of zero-th and second-order derivatives of elastic curvature fields which have to fulfil appropriate non-classical constitutive boundary and interface conditions. Relevant mesh-dependent shape functions governing the FEM technique are analytically detected. Each element is characterized by shape functions whose number is equal to four times the number of elements of the considered mesh. A simple analytical strategy to obtain nonlocal stiffness matrices and equivalent nodal forces of a finite element is exposed. The global nonlocal stiffness matrix is got by assembling the nonlocal element stiffness matrices accounting for long-range interactions among the elements. The proposed numerical approach is examined by exactly solving exemplar nonlocal case-problems of current interest in nano-engineering. The presented nonlocal strategy extends previous contributions on the matter and offers designers a consistent computational tool.

Hyperelastic structures: A review on the mechanics and biomechanics

Abstract: Soft structures are capable of undergoing reversible large strains and deformations when facing different types of loadings. Due to the limitations of linear elastic models, researchers have developed and employed different nonlinear elastic models capable of accurately modelling large deformations and strains. These models are significantly different in formulation and application. As hyperelastic strain energy density models provide researchers with a good fit for the mechanical behaviour of biological tissues, research studies on using these constitutive models together with different continuum-mechanics-based formulations have reached notable outcomes. With the improvements in biomechanical devices, in-vivo and in-vitro studies have increased significantly in the past few years which emphasises the importance of reviewing the latest works in this field. Besides, since soft structures are used for different mechanical and biomechanical applications such as prosthetics, soft robots, packaging, and wearing devices, the application of a proper hyperelastic strain energy density law in modelling the structure is of high importance. Therefore, in this review, a detailed classified analysis of the mechanics of hyperelastic structures is presented by focusing on the application of different hyperelastic strain energy density models. Previous studies on biological soft parts of the body (brain, artery, cartilage, liver, skeletal muscle, ligament, skin, tongue, heel pad and adipose tissue) are presented in detail and the hyperelastic strain energy models used for each biological tissue is discussed. Besides, the mechanics (deformation, buckling, inflation, etc.) of polymeric structures in different mechanical conditions is presented using previous studies in this field and the strength of hyperelastic strain energy density models in analysing their mechanics is presented.

Impact damage modeling in woven composites with two-level Parametrically-Upscaled Continuum Damage Mechanics Models (PUCDM)

Abstract: This paper develops two-level parametrically-upscaled continuum damage mechanics (PUCDM) models for woven composites. The PUCDM models are thermodynamically consistent, micro-/mesostructure-integrated constitutive models that bridge length scales through explicit incorporation of lower-scale descriptors into coefficients. The level-1 PUCDM model features homogenized intra-yarn constitutive response of unidirectional composite SERVEs, while the level-2 PUCDM model reflects the upscaled effect of mesostructural RVE response. The functional forms of the coefficients are established through machine learning algorithms operating on a micro-/mesostructural response database. The highly efficient two-level PUCDM models are validated using ballistic impact experiments on S-2 glass/SC-15 epoxy woven composite plates with good agreement.

General Non-Local Continuum Mechanics: Derivation of Balance Equations

Abstract: In this paper, mechanics of continuum with general form of nonlocality in space and time is considered. Some basic concepts of nonlocal continuum mechanics are discussed. General fractional calculus (GFC) and general fractional vector calculus (GFVC) are used as mathematical tools for constructing mechanics of media with general form of nonlocality in space and time. Balance equations for mass, momentum, and energy, which describe conservation laws for nonlocal continuum, are derived by using the fundamental theorems of the GFC. The general balance equation in the integral form are derived by using the second fundamental theorems of the GFC. The first fundamental theorems of GFC and the proposed fractional analogue of the Titchmarsh theorem are used to derive the differential form of general balance equations from the integral form of balance equations. Using the general fractional vector calculus, the equations of conservation of mass, momentum, and energy are also suggested for a wide class of regions and surfaces.

On nonlocal cohesive continuum mechanics and Cohesive Peridynamic Modeling (CPDM) of inelastic fracture

Abstract: In this work, we developed a bond-based cohesive peridynamics model (CPDM) and apply it to simulate inelastic fracture by using the meso-scale Xu-Needleman cohesive potential . By doing so, we have successfully developed a bond-based cohesive continuum mechanics model with intrinsic stress/strain measures as well as consistent and built-in macro-scale constitutive relations. The main novelties of this work are: (1) We have shown that the cohesive stress of the proposed nonlocal cohesive continuum mechanics model is exactly the same as the nonlocal peridynamic stress; (2) For the first time, we have applied an irreversible built-in cohesive stress-strain relation in a bond-based cohesive peridynamics to model inelastic material behaviors without prescribing phenomenological plasticity stress-strain relations; (3) The cohesive bond force possesses both axial and tangential components, and they contribute a nonlinear constitutive relation with variable Poisson's ratios; (4) The bond-based cohesive constitutive model is consistent with the cohesive fracture criterion, and (5) We have shown that the proposed method is able to model inelastic fracture and simulate ductile fracture of small scale yielding in the nonlocal cohesive continua. Several numerical examples have been presented to be compared with the finite element based continuum cohesive zone model, which shows that the proposed approach is a simple, efficient and effective method to model inelastic fracture in the nonlocal cohesive media.

Extended micropolar approach within the framework of 3M theories and variations thereof

Abstract: Abstract As part of his groundbreaking work on generalized continuum mechanics, Eringen proposed what he called 3M theories, namely the concept of micromorphic, microstretch, and micropolar materials modeling. The micromorphic approach provides the most general framework for a continuum with translational and (internal) rotational degrees of freedom (DOF), whilst the rotational DOFs of micromorphic and micropolar continua are subjected to more and more constraints. More recently, an “extended” micropolar theory has been presented by one of the authors: Eringen’s 3M theories were children of solid mechanics based on the concept of the indestructible material particle. Extended micropolar theory was formulated both ways for material systems as well as in spatial description, which is useful when describing fluid matter. The latter opens the possibility to model situations and materials with a continuum point that on the microscale consists no longer of the same elementary units during a physical process. The difference culminates in an equation for the microinertia tensor, which is no longer a kinematic identity. Rather it contains a new continuum field, namely an independent production term and, consequently, establishes a new constitutive quantity. This makes it possible to describe processes of structural change, which are difficult if not impossible to be captured within the material particle model. This paper compares the various theories and points out their communalities as well as their differences.

A Survey on Design, Actuation, Modeling, and Control of Continuum Robot

Abstract: In this paper, we describe the advances in the design, actuation, modeling, and control field of continuum robots. After decades of pioneering research, many innovative structural design and actuation methods have arisen. Untethered magnetic robots are a good example; its external actuation characteristic allows for miniaturization, and they have gotten a lot of interest from academics. Furthermore, continuum robots with proprioceptive abilities are also studied. In modeling, modeling approaches based on continuum mechanics and geometric shaping hypothesis have made significant progress after years of research. Geometric exact continuum mechanics yields apparent computing efficiency via discrete modeling when combined with numerical analytic methods such that many effective model-based control methods have been realized. In the control, closed-loop and hybrid control methods offer great accuracy and resilience of motion control when combined with sensor feedback information. On the other hand, the advancement of machine learning has made modeling and control of continuum robots easier. The data-driven modeling technique simplifies modeling and improves anti-interference and generalization abilities. This paper discusses the current development and challenges of continuum robots in the above fields and provides prospects for the future.

Effective continuum models for the buckling of non-periodic architected sheets that display quasi-mechanism behaviors

Abstract: In this work, we construct an effective continuum model for architected sheets that are composed of bulky tiles connected by slender elastic joints. Due to their mesostructure, these sheets feature quasi-mechanisms — low-energy local kinematic modes that are strongly favored over other deformations. In sheets with non-uniform mesostructure, kinematic incompatibilities arise between neighboring regions, causing out-of-plane buckling. The effective continuum model is based on a geometric analysis of the sheets’ unit cells and their energetically favorable modes of deformation. Its major feature is the construction of a strain energy that penalizes deviations from these preferred modes of deformation. The effect of non-periodicity is entirely described through the use of spatially varying geometric parameters in the model. Our simulations capture the out-of-plane buckling that occurs in non-periodic specimens and show good agreement with experiments. While we only consider one class of quasi-mechanisms, our modeling approach could be applied to a diverse set of shape-morphing systems that are of interest to the mechanics community.

Gradients and Internal Lengths in Small Scale Problems of Mechanics

Abstract: Mechanics has been a fundamental tool and principal motivation for the development of basic sciences and engineering. Its use has recently been extended from macroscopic to microscopic and nanoscopic scales and phenomena. A particular methodology in this direction is the generalization of classical theories of elasticity, plasticity and failure through the introduction of higher order gradients of the pertinent variables and corresponding internal lengths. This article, written in honor of a charismatic contributor in the field of generalized continuum mechanics, Professor Patrizia Trovalusci, begins with a simple paradigm of how to extend standard thermoelasticity theory to its gradient counterpart. It then provides a number of other examples ranging from strength of materials and stress concentrations to plastic flow and failure.

Spatial and Material Forces in Nonlinear Continuum Mechanics

Abstract: This book gives a unifying up-to-date approach to configurational mechanics and introduces material forces in a dissipation-consistent approach.

Ground state solution to the biharmonic equation

Abstract: The biharmonic equation arises in areas of continuum mechanics, including mechanics of elastic plates and the slow flow of viscous fluids. In this paper, we make an effort to establish the generalized versions of Lions-type theorem under various conditions and then apply them to study the existence of ground state solutions for the biharmonic equation.