Advanced Methods of Continuum Mechanics for Materials and Structures

This paper has many, albeit mostly didactic objectives. It is an attempt toward clarification of several concepts of continuum theory which can lead and have led to confusion. In a way the paper also creates a bridge between the lingo of the solid mechanics and the fluid mechanics communities. More specifically, an attempt will be made, first, to explain and to interpret the subtleties and the relations between the so-called material and spatial description of continuum fields. Second, the concept of time derivatives in material and spatial description will be investigated meticulously. In particular, it will be explained why and how the so-called material and total time derivatives differ and under which circumstances they turn out to be the same. To that end, material and total time derivatives will be defined separately and evaluated in context with local fields as well as during their use in integral formulations, i.e., when applied to balance equations. As a special example the mass balance is considered for closed as well as open bodies. In the same context the concept of a “moving observation point” will be introduced leading to a generalization of the usual material derivative. When the total time derivative is introduced the distinction between the purely mathematical notion of a coordinate system and the intrinsically physics-based concept of a frame of reference will gain particular importance.

Nowadays the Cosserat brothers are mostly cited for their work on so-called “Cosserat continua” of 1909 that practically initiated the theory of “oriented media” as generalized continua. But in 1896 they had already published a lengthy well-structured memoir on the theory of elasticity. This memoir is often considered as a foundational work on the modern approach to elasticity as it beautifully summarizes what was achieved in the nineteenth century but with original traits that will permeate further the twentieth century developments with an emphasis on finite deformations, the interest for applying the thermodynamic laws, the allied formulation of the notion of stress (internal forces), questions of stability, and the use of curvilinear coordinates, though still without using vector and/or tensor analysis. The present contribution examines in detail the contents of this epoch-making work of 1896, its main sources (e.g. Kirchhoff, Kelvin, Saint-Venant, Boussinesq, and Poincaré) and its insertion in the then current technical literature. We try to appraise its importance and its legacy in the modern developments of continuum mechanics, especially after the revival of the field by Truesdell and others.

A discussion of how the exiting formulas for the properties of equivalent inhomogeneities associated with Gurtin–Murdoch and spring layer models of interphases can be utilized to obtain the properties of equivalent inhomogeneities for multicomponent interphases. It is shown that in the case of energy equivalent definitions of equivalent inhomogeneities introduced recently by the present authors this can be achieved by direct superposition of the solutions associated with each component separately. General arguments are presented when such superposition is possible and it is argued that for some existing definitions of equivalent inhomogeneity this is not possible.

From the point of view of nonlinear elasticity theory the equilibrium problem for elastic sphere was considered taking into account distributed edge dislocations. We used the system of equations that consists of the incompatibility equations with a given dislocation density tensor, equilibrium equations, and constitutive equations of the material. For the isotropic material and spherically symmetric distribution of the edge dislocations, the problem was reduced to the second-order ordinary differential equation. In the framework of harmonic (semi-linear) material, the exact solution of this equation was found for any function which defines the edge dislocation density. In particular, we studied the case of dislocations concentrated on a spherical surface within a body. It was established that this surface was the discontinuity surface of strains and stresses. In addition to eigenstress problem, we solved a problem of the loading of a hollow sphere with external or internal hydrostatic pressure. Influence of dislocations on resistance of the sphere to the compression or blowing was investigated.

Following up on the classical solutions by Love for a linear-elastic self-gravitating sphere, this paper presents the corresponding extension to a linear viscoelastic body of the Kelvin–Voigt type. The solution is expressed in closed form by making use of Laplace transforms. Applications to the genesis of terrestrial planets are sought and the evolution of the Love radius and possible extensions to large deformations are discussed. As a new result, it turns out that in the early days of planet formation there is no Love radius and that it takes time for the Love radius to develop.

In industrial fabrication processes as well as in many applications of polymer parts, the glass transition plays a significant role. This is due to high mechanical processing speeds, high temperatures or large cooling rates. The mechanical, the thermomechanical and the caloric properties of polymers differ below and above the glass transition which is a thermoviscoelastic phenomenon. It depends on the ratio between the intrinsic time scale of the polymer and that of the thermomechanical loading process. If both scales are comparable, the material is in the glass transition region. Otherwise it is in the equilibrium or in the glassy region. In the industry, there are increasing demands to simulate fabrication processes in order to estimate the resulting behaviour of the polymer parts before they are manufactured. To this end, constitutive models of finite thermoviscoelasticity are needed which can represent the volumetric as well as the isochoric mechanical behaviour of the polymer in combination with the caloric and the thermomechanical properties. In a recent paper of the authors, the concept of a hybrid free energy has been developed. This approach will be applied in the current essay where the pressure-dependent relaxation behaviour under shear deformations is of interest.

In this contribution, a general formulation for constitutive equations of electromechanical active media is presented. Motivated by experimental observations, our approach is based on an additive decomposition of the Helmholtz free energy in elastic and inelastic parts and on a multiplicative decomposition of the deformation gradient in passive and active parts. The derivation is thermodynamically sound and accounts for geometric and material nonlinearities. Exemplarily, we present the solution of a uniaxial electromechanical problem and discuss the evolution of the active deformation.

A thermomechanical cyclic process of heating and cooling with fixed stresses and fixed plastic strains, respectively, at the corresponding half-cycles is discussed within the frame of large deformations and a coupled theory of Eulerian thermo-elastoplasticity. This discussion was stimulated by some comments of unknown reviewers during an in parts annoying peer-reviewing process. Within these comments, some effects observed in thermoplasticity were characterised as “against physics” and thus were used to discredit the submitted paper. In this paper, the main issues of these reviewers are re-examined for a simple uniaxial case. Accordingly, the effects are named as “some unexpected phenomena”. It could be shown that these phenomena are in accord with physics and, moreover, that comparable experimental results underline this position. The underlying Eulerian theory of thermo-elastoplasticity is associated with the recently introduced objective logarithmic rate. This paper also discloses a hitherto not published motivation for introducing this specific time derivative.

A crystal-plasticity modelling framework was implemented to simulate micromachining of a single-crystal metal. A new shear strain-based criterion was proposed to control material removal. This criterion was implemented in three different modelling techniques: element deletion, arbitrary Lagrangian–Eulerian (ALE) adaptive remeshing and smooth particle hydrodynamics (SPH) in a general-purpose finite-element software package ABAQUS. The three different modelling approaches were compared in terms of their computational accuracy and efficiency. Based on these studies, an optimized modelling strategy was proposed to simulate microscratching of single-crystal copper. The validity of the suggested methodology was corroborated through comparison between FE simulations and experimental data in terms of cutting forces, chip morphology and pile-up patterns in the work-piece.

In cyclic thermal tests of Si/solder/OFHC-Cu (silicon/solder/oxygen-free high conductivity copper)-layered plates, the authors observed either the cyclic growth or cyclic recovery of warpage to occur depending on the heat treatment of the copper before soldering. In this study, the test results are numerically analyzed by assuming three material models for the solder and two material models for the copper. It is shown that the test results are reproduced well if proper material models are used in finite element analysis. It is revealed that the so-called multiaxial ratcheting was induced in the solder, while the uniaxial type of ratcheting or cyclic strain recovery occurred in the copper. As a result, the Armstrong and Frederick model is suggested to be valid for the multiaxial ratcheting in the solder at such low strain rates as in the cyclic thermal tests, whereas the Ohno and Wang model is shown to be appropriate for the copper. To confirm this unexpected result for the solder, the Armstrong and Frederick model is applied to the multiaxial ratcheting of another solder at three strain rates.

This note deals with plastic behavior, more precisely with the role of possible additional internal variables for the description of observable effects. Based on the experimental experience, models in use are sometimes modified in order to capture new effects which cannot be described by the original version of the model. This modification can be done with additional internal variables. In doing so, one has to take care that the modified models remain thermodynamically consistent. We discuss this issue on the basis of two phenomena in cyclic plasticity. Moreover, we show how the concept of two-mechanism models can be used for the description of interactions between the arising phenomena.

It is shown that finite inelastic behavior of soft solids with the Mullins effect may be directly simulated by establishing finite elastoplastic \(J_2\)-flow models. New results in three respects are presented, including (i) general compressible deformations are taken into account for the purpose of bypassing limitations of the usual incompressibility constraint; (ii) any damage-like variables and associated evolution equations are not involved; and (iii) any given number of unloading curves of any given shapes in the Mullins effect may be simulated by direct, explicit procedures.

Effect of the geometric dimension of NiTi shape memory alloy (SMA) wires on the dissipative property of their structural components is predicted by a physical mechanism-based thermo-mechanically coupled constitutive model in this work. Two types of NiTi SMA structural components, i.e., the single-wire and multi-wire ones, are considered. The dissipative property of the component is measured by its accumulated dissipation energy obtained during cyclic deformation. The calculated results show that at low (lower than \(1\times 10^{-5}\)/s), moderate (from \(5\times 10^{-5}\)/s to \(1.5\times 10^{-4}\)/s), and high strain rates (higher than \(5\times 10^{-4}\)/s), the accumulated dissipation energy decreases, changes non-monotonically, and increases with the increasing number of wires, respectively.

Gradient-enhanced damage models find their roots in damage mechanics, which is a smeared approach from the onset, and gradients were added to restore well-posedness beyond a critical strain level. The phase-field approach to brittle fracture departs from a discontinuous description of failure, where the distribution function is regularised, which also leads to the inclusion of spatial gradients. Herein, we will consider both approaches, and discuss their similarities and differences.

This paper deals with a phenomenological damage and failure model for ductile metals. The anisotropic continuum approach takes into account the effect of stress state on damage condition and damage rule corresponding to different mechanisms acting on the micro-scale. Different branches of the criteria are formulated depending on stress intensity, stress triaxiality, and the Lode parameter. A new experimental program will be discussed in detail to validate the proposed continuum framework. Experiments with aluminum alloys are performed using a biaxial testing machine allowing individual loading of flat specimens in two directions. Loads are recorded during loading of the specimens and digital image correlation technique has been used to analyze the strain states in critical regions of the specimens. The biaxial experiments cover a wide range of stress states in shear-tension and shear-compression regimes. They will extend understanding of stress-state-dependent damage and failure mechanisms in ductile metals.

This paper was directly inspired by the presentation of Prof. Rodrigo Desmorat during The Second International Conference on Damage Mechanics, Troyes 2015 and the fruitful discussions between the first author and Profs. Holm Altenbach and Andre Dragon. The main goal of the work is not modeling the initial loading curve for concrete, but presenting the model of continuous damage deactivation able to capture subsequent unloading/reloading loops accompanying cyclic compression. A combination of conventional formulation of the universal curve of initial loading combined with Chaboche’s type damage evolution law and the continuous damage deactivation of frictional character allows for a proper quantitative and qualitative mapping of the experimental data by Sinha et al. (1964).

In this contribution, we outline the combination of a phase-field model of brittle fracture with adaptive spline-based approximations. The phase-field method provides a convenient way to model crack propagation without topological updates of the used discretisation as the crack is represented implicitly in terms of an order parameter field that can be interpreted as damage variable. For the accurate approximation of the order parameter field that may exhibit steep gradients, we utilise locally refined hierarchical B-splines in conjunction with Bézier extraction. The latter allows for the implementation of the approach in any standard finite element code. Moreover, standard procedures of adaptive finite element analysis for error estimation and marking of elements are directly applicable due to the strict use of an element viewpoint. Two different demonstration problems are considered. At first we examine the convergence properties of the phase-field approach and explain the influence of the domain size and the discretisation for the one-dimensional problem of a bar. Afterwards, results of the adaptive local refinement are compared with uniformly refined Lagrangian and spline-based discretisations. In the second example, the developed algorithms are applied to simulate crack propagation in a two-dimensional single-edge notched, shear loaded plate.

Based on the representation of the incremental stress fields by complex potentials and conformal mapping technique, the fundamental solutions for an unbounded, homogeneous, orthotropic elastic body containing an elliptical hole subjected to uniform remote loads are determined. The orthotropic body is under by uniform remote tensile, tangential, and antiplane shear loads—cases corresponding to Mode I, Mode II, and Mode III of fracture. The solutions are obtained in a compact and elementary form.

Hard foams are often used in aircraft, submarine, and automotive industry structures mostly as core in sandwich structures. The design of the critical components made from hard foams requires the knowledge of their material behaviour. Nowadays, this knowledge is gained from tests on specimens under tension, compression, torsion, and hydrostatic compression. Further tests are needed to describe the material behaviour under multi-axial loading reliably, but with default testing technology this is difficult to realize. Missing data can be predicted by numerical simulations of the microstructure. The calculated points of failure are needed to be approximated by a limit surface for the dimensioning and optimizing of engineering applications. The most known generalized strength hypotheses, however, restrict the shape of the surfaces in the principal stress space. In general, they are not suitable to describe the material behaviour of hard foams appropriately. The Capurso–Haythornthwaite generalization is chosen for the current application. It enables the description of limit surfaces with a large number of different shapes in the \(\pi \)-plane as well as varying shapes in the \(\pi \)-plane along the hydrostatic axis. The criterion takes into account the hydrostatic tensile and compressive stresses. The curvature of the meridians can be adjusted. In the current approach, a general fitting procedure is developed for the determination of the parameters of the criterion. The proposed method is not limited to polymer foams. The application to other materials like aerated concrete, cellular ceramics, and metal ceramics is possible.

The aim of the paper is to present a theoretical analysis of phenomena occurring in the 2-phase ceramic composite with the gradual degradation of the material properties under the uniaxial tension process. Ceramic composite materials have a nonlinear and complex overall response to applied loads. It is caused by the following factors: existence of an inital porosity, development of limited plasticity, different phases and internal microdefects. These microdefects cause stress concentrations and locally change the state of stress, which results in the development of mesocracks leading to macrocracks. In this contribution, a multiscale approach was applied in modelling of such material response to depict phenomena at micro- meso- and macro-scales. In experiments it was shown that defects developed mainly inter-granularly what resulted in inhomogeneity and induced anisotropy of the material.

Limitations inherent in failure theories formulated on homogenized description of composite materials are discussed. Failure mechanisms in composite materials, as understood today, are reviewed. Based on this knowledge, arguments are put forth to abandon the classical approach to formulation of failure theories for composite materials, and to instead use a computation-based failure assessment methodology. Such a methodology is proposed. In conjunction with this, the idea of virtual testing to supplement experimental determination of material response characteristics is discussed.

We consider the Cosserat continuum in its finite strain setting and discuss the dislocation density tensor as a possible alternative curvature strain measure in three-dimensional Cosserat models and in Cosserat shell models. We establish a close relationship (one-to-one correspondence) between the new shell dislocation density tensor and the bending-curvature tensor of 6-parameter shells.

Relations between plate theories resulting from the direct approach and the consistent approximation are established and the resulting equations are compared. By introducing a scalar measure for the thickness strain, both theories can be reconciled within a consistent second-order approximation.

An electromechanical coupled theory is used to develop the equations of motion of a rotating thin-walled composite beam with surface bonded/embedded piezoelectric transducers. The higher order constitutive relations for the piezoceramic material are used to take into account the impact of a high electric field. In the mathematical model of the hybrid structure, the non-classical effects like material anisotropy, rotary inertia and transverse shear deformation as well as an arbitrary beam pitch angle are incorporated. Moreover, the model considers the hub mass moment of inertia and a non-constant rotating speed case. This approach results in an additional equation of motion for the hub sub-system and enhances the generality of the formulation. It is shown that final equations of motion of the hub–beam system are mutually coupled and form a nonlinear system of partial differential equations. Comparing to the purely mechanical model, the proposed electromechanical one introduces additional stiffness-type couplings between individual degrees of freedom of the system.

The virtual work principle for two regular shell elements joined together along a part of their boundaries is proposed within the general nonlinear resultant shell theory. It is assumed that translations across the junction curve are smooth, but no restrictions are enforced on the rotations. For stiff and hinge type junctions, the curvilinear integral along the junction curve vanishes identically. In the case of deformable junction, the 1D constitutive type relation is proposed, where the constitutive function should be established by experiments for each particular engineering construction of the junction.

For both civil and mechanical engineering dynamic loads of structures are a major source of inner material damage. If (fibre) reinforced composite materials are exposed to such dynamic loads a pull-out of the reinforcing elements may occur. This dynamic pull-out of reinforcing elements is characterized by, amongst others, moving boundaries between regions of (partly) damaged and perfect bonding of reinforcement and surrounding matrix. To adequately describe these moving boundaries leads to enormous challenges. Within this contribution a simplified mechanical problem is investigated, which however provides some of the main phenomena of the dynamic pull-out. In detail, the stress and displacement fields within a rod of semi-infinite extent under a distributed load are evaluated. Herein, the front of the constant longitudinal load moves along the rod in longitudinal direction. The investigations are performed both analytically and numerically thus validating the model idealization included in the analytical solution.

This chapter is part of a research program to investigate and model the leak tightness of a Pressure Relief Valve (PRV). Presented here is: a literature review; high-temperature numerical study involving the deformation of contact faces for a metal-to-metal seal accounting for micro and macro effects; and also microscopic measurements of surface finishes and how they are modelled over a micro to nanometre scale. Currently, no review of literature exists which attempts to understand the leakage phenomenon of metal-to-metal seal contact PRV for static closed positions as they reach the set pressure point. This work attempts to do just that by drawing on inspiration from other research areas such as metal-to-metal contact and gasket seals. The key topics of interest surrounding the leakage of fluid through a gap are: fluid flow assumptions, surface characteristics and its deformation, and experimental techniques used to quantify leakage. For the numerical study, the valve geometry is simplified to an axisymmetric problem, which comprises a simple geometry consisting of only three components: a cylindrical nozzle, which is in contact with a disc (representing the valve seat on top), which is preloaded by a compressed linear spring. The nozzle-disk pair is made of the austenitic stainless steel AISI type 316N(L) steel. In a previous study, the macro–micro interaction of Fluid Pressure Penetration (FPP) was carried out in an iterative manual procedure at a temperature of 20 \(^{\circ }\)C. This procedure is now automated and implemented through an APDL script, which adjusts the spring force at a macro scale to maintain a consistent seal at elevated temperatures. Finally, using the Alicona Infinite Focus the surface form and waviness is measured, presented and modelled as 1 / 4 symmetric over a macro to nanometre scale. It is clear the surface form also needs to be accounted for, something which the literature does not focus on.

The new possibilities arisen in the last years in material manufacturing (3D-printing, electrospinning, roll-to-roll processing, self-assembly, etc.) and the theoretical tools made available by generalized continuum mechanics are still far from achieving their full potential. The main thesis of the present paper is that it is necessary a multidisciplinary approach to address the emerging issues in metamaterials’ design. Therefore, an improvement in the degree and the depth of the cooperation between scientists from different areas is required. The advancements needed in mechanics and physics of solids and fluids, mathematical and numerical modeling and advanced technology in material construction can be obtained only as a consequence of a synergic effort.

Within the framework of the model of second-gradient fluid we discuss the natural boundary conditions along edges and at corner points. As for any strain gradient model the model of second-gradient fluid demonstrates some peculiarities related with necessity of additional boundary conditions. Here using the Lagrange variational principle we derived the latter boundary conditions for various contact angles.