Material Characteristic Length Research Articles

The influence of material gradation parallel to two unequal mode-III cracks in functionally graded materials via strain gradient elasticity theory

In this article, the strain gradient elasticity (SGE) theory is used to investigate the mechanical behaviour of a functionally graded material (FGM) weakened by two unequal collinear mode-III cracks. The SGE theory uses two material characteristic lengths, ℓ and ℓ′, to account for volumetric and surface strain-gradient factors, respectively. The directions of both cracks coincide with the material gradation, which is oriented parallel to the x-axis. The numerical outcomes of the problem are obtained by constructing the system of equations using the hyper-singular integral-differential equation technique. The influence of the material gradation parameter on various fracture parameters, including the stress intensity factor (SIF), strain and crack surface displacement (CSD), is numerically shown. In addition, different loads are considered while analysing CSD profiles in connection to the gradation parameter β. Furthermore, the effect of inter-crack distance on CSD profiles is comprehensively explored, showing information between crack geometry and material gradation.

Mode-{I, III} multiaxial fatigue of welded joints in steel maritime structures: Total stress based resistance incorporating strength and mechanism contributions

Arc-welded joints in steel maritime structures are typically identified as weakest links in terms of fatigue limit state performance. Multiaxiality can be involved, consisting of predominant mode-I and non-negligible mode-III components. Aiming to answer the question if a cracked geometry based fatigue strength parameter would outperform an intact geometry based one like the effective notch stress, the total stress is adopted. A von Mises type of criterion is defined at the critical fracture plane and includes mode specific and material characteristic strength and mechanism contributions. A lifetime dependent shear strength coefficient is introduced to cover the resistance curves intercepts and slopes, whereas the total stress parameter contains the mean stress contribution as well as the (mixed) mode dependent notch and crack tip elastoplasticity coefficients, reflecting an interaction mechanism. Cycle counting includes a cycle-by-cycle non-proportionality measure and damage accumulation is based on a linear model. Evaluating mid-cycle fatigue resistance data, the total stress and effective notch stress performance turns out to be similar. However, the total stress related elastoplasticity coefficients are an explicit and sensitive measure to incorporate the actual physics of the fatigue damage process, whereas the material characteristic lengths for the effective notch stress seem to be more implicit and less sensitive ones.

Unravelling Size-Dependent and Coupled Properties in Mechanical Metamaterials: A Couple-Stress Theory Perspective.

The lack of material characteristic length scale prevents classical continuum theory (CCT) from recognizing size effect. Additionally, the even-order material property tensors associated with CCT only characterize the materials' centrosymmetric behavior and overlook their intrinsic chirality and polarity. Moreover, CCT is not reducible to 2D and 1D space without adding couples and higher-order deformation gradients. Despite several generalized continuum theories proposed over the past century to overcome the limitations of CCT, the broad application of these theories in the field of mechanical metamaterials has encountered significant challenges. These obstacles primarily arise from a limited understanding of the material coefficients associated with these theories, impeding their widespread adoption. Implementing a bottom-up approach based on augmented asymptotic homogenization, a consistent and self-sufficient effective couple-stress theory for materials with microstructures in 3D, 2D, and 1D spaces is presented. Utilizing the developed models, material properties associated with axial-twist, shear-bending, bending-twist, and double curvature bending couplings in mechanical metamaterials are characterized. The accuracy of these homogenized models is investigated by comparing them with the detailed finite element models and experiments performed on 3D-printed samples. The proposed models provide a benchmark for the rational design, classification, and manufacturing of mechanical metamaterials with programmable coupleddeformation modes.

Modeling pull-in instability of CNT nanotweezers under electrostatic and van der Waals attractions based on the nonlocal theory of elasticity

This work investigates the electromechanical response and pull-in instability of an electrostatically-actuated CNT tweezer taking into consideration a TPNL constitutive behavior of the CNTs as well as the intermolecular forces, both of which provide a significant contribution at the nanoscale. The nonlocal response of the material introduces two additional parameters in the formulation, which are effective in capturing the size effects observed at the nanoscale. The problem is governed by a nonlinear integrodifferential equation, which can be reduced to a sixth-order nonlinear ODE with two additional boundary conditions accounting for the nonlocal effects near to the CNT edges. A simplified model of the device is proposed based on the assumption of a linear or parabolic distribution of the loading acting on the CNTs. This assumption allows us to formulate the problem in terms of a linear ODE subject to two-point boundary conditions, which can be solved analytically. The results are interesting for MEMS and NEMS design. They show that strong coupling occurs between the intermolecular forces and the characteristic material lengths as smaller structure sizes are considered. Considering the influence of the nonlocal constitutive behavior and intermolecular forces in CNT tweezers will equip these devices with reliability and functional sensitivity, as required for modern engineering applications.

Mode-{I, III} multiaxial fatigue of welded joints in steel maritime structures: Effective notch stress based resistance incorporating strength and mechanism contributions

The response of maritime structures can be multiaxial, involving predominant mode-I and non-negligible mode-III components. Adopting a stress distribution formulation based effective notch stress as fatigue strength parameter for mixed mode-{I, III} multiaxial fatigue assessment purposes, a mode-I equivalent von Mises type of failure criterion has been established at the critical fracture plane. Counting includes a cycle-by-cycle non-proportionality measure and damage accumulation is based on a linear model. Distinguished mode specific and material characteristic strength and mechanism contributions in terms of respectively the resistance curve intercept and mean stress induced response ratio coefficient, resistance curve slope and material characteristic length, have been incorporated. Evaluating the mid-cycle fatigue resistance, the outperformance is impressive. The analysed multiaxial mode-{I, III} data fits the uniaxial mode-I reference data scatter band and a single resistance curve can be used for fatigue assessment.

Crack impinging on a curved weak interface: Penetration or deflection?

Curved weak interfaces present promising advantages to be implemented as crack arrestors in structures designed under the damage tolerant-design principles. Among other advantages, they neither add extra weight nor significantly affect the global stiffness of the structural element, in contrast with alternative crack arrestors concepts. To be employed as a crack arrestor, it is key that the interface is able to deviate the crack. If the crack penetrates across the interface, the effect of the weak interface as a crack arrestor is canceled. In view of this, this work studies how to set the interface parameters to promote crack deviation along the interface. In particular, following the dimensional analysis of the problem, the effect of three significant dimensionless parameters is studied: interface to bulk fracture toughness, interface to bulk tensile strength, and the interface curvature radius normalized with the material characteristic length. The corresponding analysis is carried out using three approaches widely applied for the prediction of cracking events: Linear Elastic Fracture Mechanics, Finite Fracture Mechanics, and a combination of Phase field and Cohesive Zone Model. The results present a clear effect of some parameters, such as the ratio of the interface to bulk fracture toughness, for which the three approaches agree. However, the results are moderately diverse in which correspond to the effect of the ratio of the interface to bulk tensile strength and quite divergent in what respect to the effect of the radius. The results are interpreted and explained as a consequence of the main assumptions behind the approaches studied.

Thermoelectric Joule heating in crack problems in nano-sized structures

Crack problems in nano-sized elastic structures under the Joule heating are analyzed by the finite element method (FEM). The variational principle is applied for derivation of coupled electrothermomechanical governing equations. The temperature field is described by classical Fourier′s law and it is not influenced by mechanical field. Transient heat conduction is considered since it is the slowest process among all other coupled fields here. Quasi-static conditions are considered for mechanical and electrical fields. Due to higher derivatives in governing equations the conventional FEM cannot be applied. Therefore, the mixed FEM with the C0 continuous interpolation for displacements and strains is developed. The influence of the material characteristic length on variation of crack opening displacements in finite nano-sized structures under an electric current load is investigated.

On reformulating the theory of critical distances to predict strength of notched plain concrete beams under mode I and mixed mode loading

The theory of critical distances (TCD), due to its appealing characteristics, has been successfully used in the past to predict the strength of brittle as well as ductile materials, weakened by the presence of stress risers, under both static and fatigue loading. In this work, the TCD’s unique features are exploited and the point method is reformulated to predict the strength of notched plain concrete beams of different sizes under mode I and mixed mode quasi-static loading. The fracture process zone, which is responsible for the non-linearity of concrete, is considered through the concept of an effective elastic crack. A power law is proposed to relate the effective crack length to the geometrical properties. The material characteristic length, required for application of TCD, is correlated with the maximum aggregate size. The resulting formulation is found to yield satisfactory prediction of static strength of notched plain concrete beams, wherein the geometric dimensions of the beam, tensile strength and maximum aggregate size of the concrete mix are the only input parameters. The proposed formulation is validated using a probabilistic analysis of various experimental results available in the literature.

An investigation on the evolution of strain localization zone in metallic materials based on tensile tests and a 1-D nonlocal model

Abstract Metallic materials exhibit pronounced strain localization during damage and failure, posing a challenge in damage mechanics when predicting the change in the size of the strain localization zone. In this study, uniaxial tensile tests were carried out to observe changes in the size of the strain localization zone during the loading of aluminum and low-carbon steel. The initial and final states of the two metallic materials during deformation localization were compared. The strain localization zone shrank gradually with the increase in the load, which agrees with existing electronic speckle pattern interferometry (ESPI) results. This experimental phenomenon was further analyzed theoretically. By establishing the relationship between the material characteristic length and the damage, the variation of the material characteristic length was revealed, and the form of the nonlocal kernel function with a varying characteristic length was determined. The results demonstrated that within the framework of nonlocal damage theory, the nonlocal kernel function with a varying characteristic length can be used to satisfactorily simulate the gradual shrinkage of the strain localization zone of metallic materials with the damage evolution. Therefore, this study provides an effective theoretical tool for predicting the size of the strain localization zone.

Smooth Crack Band Model—A Computational Paragon Based on Unorthodox Continuum Homogenization

Abstract The crack band model, which was shown to provide a superior computational representation of fracture of quasibrittle materials (in this journal, May 2022), still suffers from three limitations: (1) The material damage is forced to be uniform across a one-element wide band because of unrestricted strain localization instability; (2) the width of the fracture process zone is fixed as the width of a single element; and (3) cracks inclined to rectangular mesh lines are represented by a rough zig-zag damage band. Presented is a generalization that overcomes all three, by enforcing a variable multi-element width of the crack band front controlled by a material characteristic length l0. This is achieved by introducing a homogenized localization energy density that increases, after a certain threshold, as a function of an invariant of the third-order tensor of second gradient of the displacement vector, called the sprain tensorη, representing (in isotropic materials) the magnitude of its Laplacian (not expressible as a strain-gradient tensor). The continuum free energy density must be augmented by additional sprain energy Φ(l0η), which affects only the postpeak softening damage. In finite element discretization, the localization resistance is effected by applying triplets of self-equilibrated in-plane nodal forces, which follow as partial derivatives of Φ(l0η). The force triplets enforce a variable multi-element crack band width. The damage distribution across the fracture process zone is non-uniform but smoothed. The standard boundary conditions of the finite element method apply. Numerical simulations document that the crack band propagates through regular rectangular meshes with virtually no directional bias.

Magnetohydrodynamic Micropolar Fluid Squeeze Film Lubrication between Stepped Porous Parallel Plates

Objectives: The primary goal of this paper is to study the influence of MHD and micropolar fluid on the squeeze film lubrication between stepped porous parallel plates. Method: The non-Newtonian micropolar fluid MHD Reynolds type equation is derived by considering the flow of micropolar fluid in the porous matrix as described by Darcy’s law, as well as microstructure additives and magnetic effects associated with the magnetization of the fluid. The numerical solutions are presented graphically for the MHD squeeze film characteristics for various values of coupling number parameter, characteristic material length, and magnetic Hartmann number. Findings: According to the results, the micropolar fluid and the magnetic effects significantly influence the squeeze film characteristics. Comparing the MHD micropolar fluid impact on the squeeze film lubrication with the corresponding Newtonian and non-magnetic cases, we observe that there is a significant increase in the approaching time and the load-carrying capability. The increase in the step height decreases the squeezing film time. Novelty: The original research was conducted on the magneto-hydrodynamic micropolar fluid squeeze film lubrication between stepped porous parallel plates which has not been studied so far. The effect of applied magnetic field is to enhance the load carrying capacity and delayed time of approach which are the most desirable characteristics for improving the bearing performance. Keywords: Squeeze Film; Stepped plates; Magnetohydrodynamic; Porous; Micropolar

Interaction of a semi-infinite crack with a screw dislocation within Mindlin’s first strain-gradient elasticity

Behavior of dislocations in the vicinity of cracks and the stresses induced in a material due to their interactions have a significant influence on the toughness of that material. As a fundamental problem related to this issue, the present paper is hence devoted to address the interaction of a semi-infinite crack and a straight screw dislocation in an infinite isotropic material within the framework of Mindlin’s first strain-gradient theory. Analytical expressions are determined for the induced displacement and elastic strain fields as well as the image force acting on the dislocation core. Subsequently, the limiting behaviors of the displacement and elastic strain fields in the vicinity of the crack tip are investigated. The current analysis shows that the solution obtained for the displacement field can be represented by a smooth function everywhere in the medium, especially on the slip plane of the dislocation and around the crack tip. Moreover, it is revealed that, according to the adopted theory, all classical singularities of the elastic strain field at the crack tip and dislocation core are eliminated. The obtained results also exhibit clearly the size effect on the induced displacement and elastic strain fields, in particular when the geometrical and material characteristic lengths of the problem are comparable to one another.

Indentation of a free beam resting on an elastic substrate with an internal lengthscale

The plane strain problem of a slender and weightless beam-plate loaded by a transversal point force in unilateral contact with a couple stress elastic foundation is investigated. The study aims to explore the consequences of the material internal lengthscale on the contact mechanics. In particular, compatibility between the beam and the foundation surface demands that both displacement and rotation match along the contact line. To this aim, couple tractions are exchanged besides the traditional contact pressure until separation between the beam and the foundation occurs. The problem is formulated making use of the Green's functions for a point force and a point couple acting atop of a couple stress elastic half-plane. A pair of coupled integral equations is thus derived, that governs the distribution of contact pressure and couple tractions, with one of them being immediately solved to provide an explicit relation between the two unknowns. In this sense, we retrieve the concept of a mechanically equivalent action, as it is the case of the Kirchhoff shear for plates. The remaining integral equation sets a cubic eigenvalue problem, whose linear term accounts for the microstructure. Its numerical solution is sought by expanding the equivalent contact pressure in series of Chebyshev polynomials vanishing at the contact region ends points, namely the lift-off points, and then applying a collocation strategy. The contact length, the distributions of contact pressure and couple tractions under the beam and the shearing force and bending moment along the beam are then obtained as a function of the material characteristic length. Results clearly indicate that accounting for the material internal lengthscale is mainly realized through exchange of the couple tractions, in the lack of which results much resemble those of the classical solution. Specifically, greater contact lengths and a stronger focusing effect about the loading point are encountered, which become very significant when the contact length approaches the internal lengthscale.

Investigating the effect of notch size on critical distance and fatigue limit by coupling the theory of critical distance and finite fracture mechanics

The energy based failure criterion Finite Fracture Mechanics is transformed into a stress based failure criterion and expressed by an effective stress similar with Line Method. After coupling the two criterions, the theoretical critical distance expression is derived based on notch stress distribution, stress intensity factor function and material characteristic length. The effect of notch size and material characteristic length on the critical distance and fatigue limit prediction was analyzed for a group of V-shaped notches. Then, the coupled method was validated by a large number of experimental results. It is revealed that the Line Method and coupled method are the same except that they employ different critical distances. The validity of Line Method was discussed based on the difference of critical distances and fatigue limit predictions between the Line Method and coupled method.

Mode-III fatigue of welded joints in steel maritime structures: Weld notch shear stress distributions and effective notch stress based resistance

The predominant mode-I response of maritime structures can be multiaxial, involving out-of-plane mode-III shear components. Semi-analytical mode-III notch stress distribution formulations have been established for critical details like welded T-joints and cruciform joints, reflecting (non-)symmetry with respect to half the plate thickness. Using a stress distribution formulation based effective notch stress as fatigue strength criterion, the mode-III welded joint mid-cycle fatigue resistance characteristics have been investigated. In comparison to mode-I, the material characteristic length and resistance curve slope estimate suggest the fatigue damage process to be even more an initiation related near-surface phenomenon. Mean shear stress effects seem insignificant.

Axisymmetric adhesive contact of multi-layer couple-stress elastic structures involving graded nanostructured materials

Graded nanostructured (GNS) materials has attracted widespread attention in the recent years. The gradually changed microstructures and mechanical properties from the surface to the interior layer in graded nanostructured materials can improve the mechanical properties of the surface. The motivation of this research was to explore the effect of size effect, adhesion and gradient on the mechanical properties in contact between GNS coating and spherical indenter. The multi-layer couple-stress elastic structures is used to simulate the GNS coating with arbitrarily varying material properties. Based on the Maugis-Dugdale theory, the axisymmeric adhesive contact problem for multi-layer couple-stress elastic coated substrate system indented by spherical punch is converted to singular integral equations. Numerically calculation are conducted to obtain the influence of the characteristic material length, adhesion parameter and gradient index on the indentation, surface stress, surface displacement, total attraction force. The present results should be helpful for the understanding of the micro contact mechanics of GNS coating.

Lattice infill structure design of topology optimization considering size effect

The vigorous development of additive manufacturing technology promotes the tendency of lattice infill design to be miniaturized. The representative scale of the deformation field becomes equivalent to the length scale of the microstructures when the characteristics size of the lattice infill design is down to micron or nanoscale, and strain gradient effects or size effects arise in the structure. Due to the lack of microscopic material properties, the typical topology optimization framework based on classical mechanics for lattice infill structure cannot represent the performance difference induced by the size effect. Therefore, single-scale lattice infill structure design and couple stress theory are combined to explore the size effect on topology optimization. Numerical examples show that when the size ratio of the macroscopic feature size to the material characteristic length is reduced to a comparable range, the topological configurations and the target compliance values exhibit a significant size effect.

Free transverse vibrations of nanobeams with multiple cracks

A nonlocal model is formulated to study the size-dependent free transverse vibrations of nanobeams with arbitrary numbers of cracks. The effect of the crack is modeled by introducing discontinuities in the slope and transverse displacement at the cracked cross-section, proportional to the bending moment and the shear force transmitted through it. The local compliance of each crack is related to its stress intensity factors assuming that the crack tip stress field is undisturbed (non-interacting cracks).The kinematic field is defined based on the Bernoulli-Euler beam theory, and the small-scale size effect is taken into account by employing the constitutive equation of the stress-driven nonlocal theory of elasticity. In this manner, the curvature at each cross-section is defined as an integral convolution in terms of the bending moments at all the cross-sections and a kernel function which depends on a material characteristic length parameter. The integral form of the nonlocal constitutive equation is elaborated and converted into a differential equation subjected to a set of mathematically consistent boundary and continuity conditions at the nanobeam's ends and the cracked cross-sections. The equation of motion in each segment of the nanobeam between cracks is solved separately and the variationally consistent and constitutive boundary and continuity conditions are imposed to determine the natural frequencies. The model is applied to nanobeams with different boundary conditions and the natural frequencies and the mode shapes are presented at the presence of one to four cracks. The results of the model converge to the experimental results available in the literature for the local cracked beams and to the solutions of the intact nanobeams when the crack length goes to zero. The effects of the crack location, crack length, and nonlocality on the natural frequencies are investigated, also for the higher modes of vibrations. Novel findings including the amplification and shielding effects of the cracks on the natural frequencies are presented and discussed.

A two characteristic length nonlocal GTN model: Application to cup–cone and slant fracture

Ductile failure prediction is essential to avoid loss of structural integrity or to control crack propagation during forming processes. One of the main difficulties is the prediction of complex crack paths. This study proposes a nonlocal extension of a local Gurson–Tvergaard–Needleman (GTN) ductile damage model at finite strains able to capture cup–cone or slant fracture. The proposed model is based on an implicit gradient formulation which enables to solve the problem of spurious strain and damage localization. The model integrates two different material characteristic lengths, which are used to separately regularize damage by void growth and damage by damage nucleation. After parameter fitting using data for a line pipe steel, conditions to obtain converged solutions are studied, which can be used to select the mesh size in the localization band. With the appropriate mesh design, it was possible to study the effect of the characteristic lengths on the formation of cup–cone fracture and slant fracture. It is observed that, for a given specimen size, larger characteristic lengths favor flat crack advance. Similarly, for given material lengths, size effects can be predicted with small specimens being more prone to flat fracture. This paves the way to a more direct determination of material lengths by using homothetic specimens so as to obtain different crack paths.

Regularization of localization due to material softening using a nonlocal hardening variable in an Eulerian formulation of inelasticity

It is known that damage or inelastic softening can cause an ill-posed problem leading to localization and mesh-dependence in finite element simulations. In this paper, a nonlocal hardening variable, κ̄, is introduced in a finite deformation Eulerian formulation of inelasticity with a rate-independent smooth elastic–inelastic transition. This nonlocal variable is defined over an Eulerian region of nonlocality, which is a sphere with radius equal to the characteristic length, Lc, defined in the current deformed geometry of the material. Two models of this nonlocal hardening variable are explored. One model where κ̄ is the minimum value of the local hardening κ within the region of nonlocality, and another model where κ̄ is the average of κ in the same region. The influence of the nonlocal hardening variable is studied using an example of a plate that is loaded by a prescribed boundary displacement causing formation of a shear band. Predictions of the applied load vs. displacement curves and contour plots of the total distortional deformation of the plate and the hardening variable κ are studied. The model based on the minimum value of κ in the nonlocal region predicts mesh-independent post-peak response of the load vs. displacement curve. Also, it is shown that the characteristic material length, Lc, controls the structure of the shear band developed in the plate.