Displacement Field

This paper represents a further contribution to the study of identification procedures for material mechanics resting on kinematic measurements provided by 2D Digital Image Correlation (DIC) at the microscale. Reference is made to non-conventional experiments on adhesively bonded assemblies industrially manufactured for aerospace applications. For calibration purposes a local approach is considered under plane stress conditions, focusing on a small sub-domain on the sample surface, in which mixed mode debonding is monitored. As a novelty, both the (cohesive) mechanical parameters of the interface and the actual boundary conditions prescribed at different time instants during the test are considered as unknowns to be estimated on the basis of full-field data. In this way, data smoothing and parameter identification procedures, so far usually performed in a sequence, are tackled simultaneously in a coupled framework. Since the inverse problem generalized as mentioned above turns out to be severely ill-posed, suitable regularizing provisions are applied, concerning the a priori regularity of (kinematic) displacement fields, from which boundary data are sampled, and the equilibrium (Neumann) conditions along the cracked part of the interface.

An experimental-numerical methodology is introduced to identify the parameters of a cohesive law of an adhesive layer within a joined assembly on the basis of kinematic data provided by digital image correlation. Nonconventional experiments on joined samples were designed to generate within the assembly and the adhesive film complex strain and stress states close to those expected in service and up to complete debonding. The modeling is developed with reference to the observed sub-domain in which the experimental boundary conditions are prescribed. The non-linear behavior of the adhesive layer is described as a finite thickness interface endowed with a mixed-mode cohesive law whose parameters are identified so as to match at best the measured displacement field.

Although exact solutions for linear static analysis of most frame structures can be obtained by the finite element method, it is very difficult to get exact solutions for free vibration and harmonic analyses for nontrivial cases. This paper extends an earlier study on exact solutions of axial vibration problems of elastic bars to dynamic analyses of elastic frame structures. New shape functions for the transverse displacement field are constructed by using the homogeneous governing equations and then a novel beam element is formulated. Combining the new (bending) beam element and the one developed earlier for elastic bars yields a new element for general frame structures. The new frame element can be used to get exact solutions for both natural frequency and undamped harmonic analyses of frame structures. Illustrative examples are presented to demonstrate the effectiveness of the new element and the algorithm. Copyright

This paper proposes a new finite-element modelling of a recent layerwise model for multilayered plates. This layerwise model is built from a specific 3D stress-field expansion along the thickness direction and involves, in particular, interlaminar transverse shear and out-of-plane stresses as generalized stresses. Its main feature is that 3D equilibrium equations and free-edge boundary conditions are directly taken into account into the stress-based construction of the model. A dual displacement-based finite-element discretization is implemented using the FEniCS software package and a remeshing strategy is proposed based on a novel error indicator. The error indicator is built based on the 3D stress field directly deduced from the layerwise generalized stresses and compared to a reconstructed stress field based on the model generalized displacements. The proposed error indicator is shown to identify the most critical parts of a laminate structure associated with complex 3D stress fields such as boundaries or stress concentration/singularity regions (near free-edges or delamination fronts). Through the combination of thickness discretization and in-plane mesh refinement in regions of interest, the proposed framework therefore offers an attractive alternative to 3D solid finite elements for an accurate prediction of stress states in composite laminates.

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In this paper, an elastic, rectangular, and simply supported, sigmoid functionally graded material (S-FGM) beam of thick thickness subjected to uniformly distributed transverse loading has been investigated. The S-FGM system consists of ceramic (Al 2 O 3) and metal (Al) phases varying through the thickness of beam. Major classes of representative theories such as classical laminate beam theory (CLBT), first-order shear deformation theory (FSDT) and high-order theories (HOTs) have been considered and a unified kinematic formulation is then proposed. The Poisson's ratio of the thick S-FGM beam is assumed to be constant, but their Young's moduli vary continuously throughout the thickness direction according to the volume fraction of constituents defined by sigmoid function. The numerical illustrations concern bending response of S-FGM rectangular beams. Qualitative and quantitative assessments of displacement and stress fields have been presented and discussed.

It is well known that classical continuum theory has certain deficiencies in capturing the size effects and predicting the nanoscopic behavior of materials in the vicinity of nano-inhomogeneities and nano-defects with reasonable accuracy. Couple stress theory which is associated with an internal length scale for the medium is one of the higher order continuum theories capable of overcoming such difficulties. In this work, the problem of a nano-size fiber embedded in an unbounded isotropic elastic body for three different types of interface conditions: perfect, imperfect (partially damaged), and pure sliding (completely damaged) subjected to remote anti-plane loading is examined in this framework. The physically realistic size-dependent elastic fields for the problem will be derived analytically. The discontinuities of the displacement and rotation fields at the imperfect interfaces are assumed to be proportional to the associated reduced traction and couple traction, respectively. The effect of the interfacial damage on the stress field around the nano-fiber is also examined. Subsequently, the elastic field of a single nano-fiber with a damaged interface condition is employed in conjunction with the Mori-Tanaka theory to estimate the size-dependent overall anti-plane shear modulus of such solids enriched with unidirectional circular cylindrical fibers severely damaged at their interfaces with the matrix. The dependence of the anti-plane elastic shear modulus on several important physical parameters such as size, interface conditions, rigidity of the fiber, and the characteristic length of the constituents is analyzed. Finally, a variational approach for the estimation of the upper and lower bounds of anti-plane shear modulus will be given within couple stress elasticity and, moreover, the dependence of the bounds on the matrix-fiber interface damage and the fiber to the matrix rigidity ratio is examined.

It is well known that classical continuum theory has certain deficiencies in capturing the size effects and predicting the nanoscopic behavior of materials in the vicinity of nano-inhomogeneities and nano-defects with reasonable accuracy. Couple stress theory which is associated with an internal length scale for the medium is one of the higher order continuum theories capable of overcoming such difficulties. In this work, the problem of a nano-size fiber embedded in an unbounded isotropic elastic body for three different types of interface conditions: perfect, imperfect (partially damaged), and pure sliding (completely damaged) subjected to remote anti-plane loading is examined in this framework. The physically realistic size-dependent elastic fields for the problem will be derived analytically. The discontinuities of the displacement and rotation fields at the imperfect interfaces are assumed to be proportional to the associated reduced traction and couple traction, respectively. T...

The demand for designing lightweight structures without any loss of strength or stiffness has conducted many engineers and researchers to seek for alternative joining methods. In this context, adhesive bonding may appear as an attractive joining method. However the interest of adhesive bonding remains while the structural integrity of the joint is ensured. According to recent literature the Cohesive Zone Model (CZM) appears as a suitable approach able to predict both static and fatigue strength of adhesively bonded joints. This approach of the fracture process of adhesive layers is based on the modeling of the adhesive mechanical behavior through a set of adhesive cohesive properties in either mode I, mode II or mixed-mode I/II. The strength prediction of adhesively bonded joints is then highly dependent on the CZM parameters. The methods used to experimentally characterize them are thus essential. A new methodology, termed direct method, is presented and tested. It is based on the measurement of displacement field of bonded adherends at the crack tip of classical specimens allowing for the loading of the adhesive layer in pure mode I, pure mode II and mixed-mode I/II. The tested adhesive is a methacrylate-based two-component adhesive paste found under the reference SAF30 MIB manufactured by AEC Polymers (ARKEMA group). The adherends are in aluminium 6060. It is shown that it is possible to characterize the cohesive properties of the adhesive layer using the direct method. The numerical tests involve both adherends and adhesive nonlinearities. Nevertheless, the presented experimental implementation passes by the development of a dedicated data pre-processing to interpret the experimental measurements, highlighting the significance of the choice of the measurement means linked to the design of specimen.

Adhesive bonding is usually modelled using cohesive zone models (CZM) which are defined by traction-separation (TS) law. For mode I loading condition these phenomenological laws simply represent the evolution of the peel stress as a function of the two adherends relative displacement normal to the joint. However, TS law shape is often empirically chosen rather than being measured. The uncertainty on parameter estimation is generally not indicated even though it strongly influences the reliability of the bonded joint strength prediction. Moreover there are several mechanical data that can be obtained experimentally from crack initiation and propagation experiments on a Double Cantilever Beam Test (DCB). In general, TS parameters are chosen from load-displacement curves, which is the most straightforward mechanical response to obtain. However, the development of digital image correlation has enabled to access more numerous data, such as adherends’ deflection and rotation along the ove...

The demand for designing lightweight structures without any loss of strength or stiffness has conducted many engineers and researchers to seek for alternative joining methods. In this context, adhesive bonding may appear as an attractive joining method. However the interest of adhesive bonding remains while the structural integrity of the joint is ensured. According to recent literature the cohesive zone model (CZM) appears as a suitable approach able to predict both static and fatigue strength of adhesively bonded joints. This approach of the fracture process of adhesive layers is based on the modeling of the adhesive mechanical behavior through a set of adhesive cohesive properties in either mode I, mode II or mixed-mode I/II. The strength prediction of adhesively bonded joints is then highly dependent on the CZM parameters. The methods used to experimentally characterize them are thus essential. A new methodology, termed direct method, is presented and tested. It is based on the measurement of displacement field of bonded adherends at the crack tip of classical specimens allowing for the loading of the adhesive layer in pure mode I, pure mode II and mixed-mode I/II. The tested adhesive is a methacrylate-based two-component adhesive paste found under the reference SAF30 MIB manufactured by AEC Polymers (ARKEMA group). The adherends are in aluminium 6060. It is shown that it is possible to characterize the cohesive properties of the adhesive layer using the direct method. The numerical tests involve both adherends and adhesive nonlinearities. Nevertheless, the presented experimental implementation passes by the development of a dedicated data pre-processing to interpret the experimental measurements, highlighting the significance of the choice of the measurement means linked to the design of specimen.

Background: Residual stresses often cause distortion to occur when machining slender workpieces like those used in aeronautics. This distortion is undesirable and may even cause the workpiece to be scrapped. Identifying these residual stresses during machining is therefore crucial to limit their effect on the final geometry of the part. Objective: The aim of this work is to identify the through-thickness residual stress distribution by processing displacement/strain fields measured on the workpiece during machining. Methods: Digital Image Correlation is employed to measure, between successive milling passes, the displacement and strain fields on the lateral surface of the workpiece. These fields are then processed with the Virtual Fields Method to identify the through-thickness residual stress distribution. Compared to previous studies on this topic, no assumption is made concerning the real through-thickness displacement field. 1 Results: Simulations performed with synthetic data provided by a finite element model show the feasibility of this approach and quantifies its robustness when displacements are affected by measurement noise. Results obtained with this approach in a real case are then compared with their counterparts obtained with another identification technique, which assumes that the workpiece behaves as a beam. Conclusion: This study shows that it is possible to measure the through-thickness distribution of residual stress in slender workpieces during machining by using a classic subset-based DIC software and the Virtual Fields Method to process the displacement/strain maps. This opens the way for future developments aimed for instance at updating in live machining sequences in order to obtain machined parts immune from distortions caused by residual stresses.

We present thc digital image correlation method (DICM) using the surface configuration data by the scanning probe mjcrr) scope (SPM) ． The bending test is carried out on the table ofthe scanning probe microscope ， and surface configuration data obtained by the SPM is employed in computation of displacement field by the DICM ． In addition ， afier the ◎omputation ofthe displacement field ， the moVing least square method is applied toOcompUte the strain field ．

An efficient C 0 continuous two dimensional (2D) finite element (FE) model is developed based on a refined higher order shear deformation theory (RHSDT) for the static analysis of soft core sandwich plate having imperfections at the layer interfaces. In this (RHSDT) theory, the in-plane displacement field for the face sheets and the core is obtained by superposing a globally varying cubic displacement field on a zig-zag linearly varying displacement field. The transverse displacement is assumed to have a quadratic variation within the core and it remains constant in the faces beyond the core. In this theory, the interfacial imperfection is represented by a liner springlayer model. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C 1 continuity requir...

An efficient C0 continuous finite element (FE) model is developed based on a combined theory (refine higher order shear deformation theory (RHSDT) and least square error (LSE) method) for the static analysis of a soft core sandwich plate. In this (RHSDT) theory, the in-plane displacement field for the face sheets and the core is obtained by superposing a global cubically varying displacement field on a zig-zag linearly varying displacement field with a different slope in each layer. The transverse displacement assumes to have a quadratic variation within the core and it remains constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the sandwich plate. The nodal field variables are chosen in an efficient manner to circumvent the problem of C1 continuity requirement of the transverse displacements. In order to calculate the ac...

This paper deals with the methodology in the processing of airborne SAR data to measure glacier displacement fields. The possibility to retrieve a 2D displacement map of the deformation in slant-range geometry with an airborne platform is discussed. A new extended multisquint approach is proposed to simultaneously estimate residual motion errors and the along-track displacement of the glacier, while the across-track displacement is obtained by means of differential interferomatry. Experimental results are shown with data acquired by the Experimental SAR (E-SAR) of the German Aerospace Center over the Aletsch glacier in the Swiss Alps.

Mechanical compression and tearing tests are carried out on crimped glass wool samples. The displacement field is determined by using digital image correlation based on images taken at different stages of the mechanical tests. A multiscale algorithm is used to resolve accurately fine details of the displacement field. This technique reveals strain heterogeneities and further localisation in compression tests well below the peak stress. Crack formations are identified in tearing tests. Reliability and resolution of the displacement and strain fields are validated by using different window sizes in the correlation analysis.

Mechanical compression and tearing tests are carried out on crimped glass wool samples. The displacement field is determined by using digital image correlation based on images taken at different stages of the mechanical tests. A multi-scale algorithm is used to resolve accurately the fine details of the displacement field. This technique reveals strain heterogeneities and further localization in compression tests well below the peak stress. Crack formations are identified in tearing tests. Reliability and resolution of the displacement and strain fields are validated by using different window sizes in the correlation analysis.

The final aim of this work is to build a tool dedicated to the calculation of the mechanical fields in pavements incorporating possible vertical cracks in some layers or partial debonding at the interface between layers. The development of this tool is based on a specific layer-wise modeling of the structure so-called M4-5n. In this model the stress fields are approached through polynomial approximations in the vertical direction for each layer. Its construction is based on the Hellinger-Reissner (H-R) variational principle of continuum mechanics. One advantage of the M4-5n is to reduce by one the dimension of the problem. Moreover this model leads to finite values of the generalized interface stresses at the crack lips of the structures studied. This approach is thus particularly adapted to parametric studies and might be considered for analyzing crack growth in layered structures such as pavements. The contribution of the present paper to this model is focused on the computation of its numerical solution by means of the mixed Finite Element Method (FEM). The developed method is based on the maximum of the complementary energy theorem using Lagrangian multipliers to ensure the equilibrium equations. The resulting formulation is equivalent to the H-R variational principle applied to the generalized displacement and stress fields. This approach is applied to a beam structure composed of four elastic homogenous layers resting on Winkler's springs. Vertical cracks across some layers are introduced. The results obtained are compared with those from an earlier approach using the Finite Difference Method (FDM).

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors. IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the von-Karman strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.

While a lot is known about the deformation of metallic surfaces from experiments, elasticity theory and simulations, this investigation represents the first molecular-dynamics-based simulation of uniaxial deformation for the vicinal surfaces in a comparison of copper and nickel. These vicinal surfaces are composed of terraces divided by equidistant, mono-atomic steps. The periodicity of vicinals makes them good candidates for the study of the surface steps’ influences on surface dynamics. The simulations of tensile and compressive uniaxial deformations were performed for the (1 1 19) vicinal surfaces. Since the steps on the surfaces serve as stress concentrators, the first defects were expected to nucleate here. In the case of copper, this was found to be the case. In the case of nickel, however, dislocations nucleated beneath the near-surface layer affected by the displacement field generated by the steps. Slip was hindered at the surface step by the vortex in the displacement fiel...

This paper deals with the mechanical characterisation of elastomeric materials. An original method is proposed to identity the material parameters. It consists of performing only one heterogeneous mechanical test, measuring the displacement/strain field using suitable Digital Image Correlation software and applying an inverse method, namely the Virtual Fields Method, to process the resulting displacement/strain maps. For this purpose, a new apparatus is designed to be adapted to a conventional tensile machine. This apparatus enables us to obtain simultaneously uniaxial tension, pure shear and equibiaxial tension, using only one sample. The heterogeneity of the kinematic fields induced by the test is first discussed in relation to two criteria. The main features of the identification method are then presented, and results provided by a test performed on an elastomeric material are discussed in the context of hyperelasticity. Keywords test-induced heterogeneity, hyperelasticity, inverse problem, identification P is the transition matrix between the Cauchy stress tensor expressed in any 1-2 basis and its eigenbasis.

Design and advancement of the durable urban train infrastructures are of utmost importance for reliable mobility in the smart cities of the future. Given the importance of urban train lines, tunnels, and subway stations, these structures should be meticulously analyzed. In this research, two-dimensional modeling and analysis of the soil-structure mass of the Alan Dasht station of Mashhad Urban Train are studied. The two-dimensional modeling was conducted using Hashash’s method and displacement interaction. After calculating the free-field resonance and side distortion of the soil mass, this resonance was entered into PLAXIS finite element program, and finally, stress and displacement contours together with the bending moment, shear force and axial force curves of the structure were obtained.

We first establish that the linearized change of metric and change of curvature tensors, with components in L2 and H-1 respectively, associated with a displacement field, with components in H1, of a surface S immersed in ℝ3 must satisfy in the distributional sense compatibility conditions that may be viewed as the linear version of the Gauss and Codazzi-Mainardi equations. These compatibility conditions, which are analogous to the familiar Saint Venant equations in three-dimensional elasticity, constitute the Saint Venant equations on the surface S. We next show that these compatibility conditions are also sufficient, i.e. that they in fact characterize the linearized change of metric and the linearized change of curvature tensors in the following sense: If two symmetric matrix fields of order two defined over a simply-connected surface S ⊂ ℝ3 satisfy the above compatibility conditions, then they are the linearized change of metric and linearized change of curvature tensors associat...

If a symmetric matrix field e of order three satisfies the Saint Venant compatibility conditions in a simply-connected domain Ω in R 3 , there then exists a displacement field u of Ω such that e = 1 2 (∇u T + ∇u) in Ω. If the field e is sufficiently smooth, the displacement u(x) at any point x ∈ Ω can be explicitly computed as a function of e and CURL e by means of a Cesàro-Volterra path integral formula inside Ω with endpoint x. We assume here that the components of the field e are only in L 2 (Ω), in which case the classical path integral formula of Cesàro and Volterra becomes meaningless. We then establish the existence of a "Cesàro-Volterra formula with little regularity", which again provides an explicit solution u to the equation e = 1 2

Let ω be a simply-connected open subset in R 2 and let θ : ω → R 3 be a smooth immersion. If two symmetric matrix fields (γ αβ) and (ρ αβ) of order two satisfy appropriate compatibility relations in ω, then (γ αβ) and (ρ αβ) are the linearized change of metric and change of curvature tensor fields corresponding to a displacement vector field η of the surface θ(ω). We show here that, when the fields (γ αβ) and (ρ αβ) are smooth, the displacement vector η(y) at any point θ(y), y ∈ ω, of the surface θ(ω) can be explicitly computed by means of a "Cesàro-Volterra path integral formula on a surface", i.e., a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γ αβ and ρ αβ and their covariant derivatives.

The electronic response in solids is considered as a displacement and a deformation of the electron liquid. The electronic displacement field is determined from the self-consistent equation following from the minimum energy condition. It is shown that this technique is convenient for calculations ofthe elastic properties.

This paper presents a new method for calculating the elastic stress intensity factors for 3D structural components under with complex loads and geometry. To this end a hybrid element formulation is developed where the hybrid element has its stiffness matrix corrected to exactly reflect the existence of the true 3D crack geometry within the element. To obtain the stiffness matrix, of the hybrid element, this approach involves the combined use of the redundant force method together with the displacement field results arising from the finite element alternating technique (FEAT). This matrix was subsequently used together with a standard commercial finite element package to obtain the nodal force. These forces were then used to determine the stress intensity factors along the crack front. This procedure was validated by comparison with results available in literature.

This is the first study on the free vibrational behavior of sandwich panels with flexible core in the presence of smart sheets of oil which is capable of the excitation of magnetic field. In order to model the core, the improved high order theory of sandwich sheets was used by a polynomial with unknown coefficients first degree shear theory was used for the sheets. The derived equations based on Hamilton principle with simple support boundary condition for upper and lower sheets were solved using Navier technique. Accuracy and precision of the theory were investigated by comparing the results of this study with those of analytical and numerical works. In the conclusion section, effect of the intensity of magnetic field and other physical parameters including ratio of sheet's length to width, ratio of sheet's length to thickness, ratio of core thickness to sheet's overall thickness, and ratio of oil layer thickness to sheet's overall thickness on natural frequency was...

SUMMARYIn this paper, a phase‐field model for cohesive fracture is developed. After casting the cohesive zone approach in an energetic framework, which is suitable for incorporation in phase‐field approaches, the phase‐field approach to brittle fracture is recapitulated. The approximation to the Dirac function is discussed with particular emphasis on the Dirichlet boundary conditions that arise in the phase‐field approximation. The accuracy of the discretisation of the phase field, including the sensitivity to the parameter that balances the field and the boundary contributions, is assessed at the hand of a simple example. The relation to gradient‐enhanced damage models is highlighted, and some comments on the similarities and the differences between phase‐field approaches to fracture and gradient‐damage models are made. A phase‐field representation for cohesive fracture is elaborated, starting from the aforementioned energetic framework. The strong as well as the weak formats are p...

Full displacement and stress ®elds in a two-dimensional elastic solid with multiple interacting cracks is constructed in an approximate way, in elementary functions. It is done by approximating the actual displacement discontinuity (COD) on each crack, in each of modes I and II, by a product of the ellipse corresponding to an isolated crack and a quadratic polynomial. The polynomial's coecients, that re¯ect crack interactions, are chosen to match the SIFs and the average tractions on cracks (quantities found, for example, by a simple method of analysis of crack interactions; Kachanov, M. Elastic solids with many cracks: a simple method of analysis. International Journal of Solids and Structures, 23, 1987. 23±43). We then use this approximated CODs in the representation of the displacement ®eld in the solid in terms of integrals over crack lines, with CODs as kernels. The constructed ®eld diers from the actual one by the ®eld generated by such traction distributions on cracks that have zero averages and produce zero stress intensity factors. The accuracy of the suggested method depends on the con®guration, but, generally, remains good at spacings between cracks that are substantially smaller than the crack lengths.

In this paper we deal with the inverse problem of determining cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement’s measurements. For, we consider a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term that penalizes the perimeter of the cavity or inclusion to be identified. Then using a phase field approach we derive a robust algorithm for the reconstruction of elastic inclusions and of cavities modelled as inclusions with a very small elasticity tensor.

In this paper, we investigate a rate‐independent model for hybrid laminates described by a damage phase‐field approach on two layers coupled with a cohesive law governing the behaviour of their interface in a one‐dimensional set‐up. For the analysis, we adopt the notion of energetic evolution, based on global minimisation of the involved energy. Due to the presence of the cohesive zone, as already emerged in literature, compactness issues lead to the introduction of a fictitious variable replacing the physical one which represents the maximal opening of the interface displacement discontinuity reached during the evolution. A new strategy which allows to recover the equivalence between the fictitious and the real variable under general loading–unloading regimes is illustrated. The argument is based on time regularity of energetic evolutions. This regularity is achieved by means of a careful balance between the convexity of the elastic energy of the layers and the natural concavity of...