Special virtual fields for the direct determination of material parameters with the virtual fields method. 3. Application to the bending rigidities of anisotropic plates

This paper deals with the direct identification of bending rigidities of thin anisotropic plates. These parameters are extracted from an heterogeneous strain field which takes place onto the top surface of a bent plate. The loading conditions are such that no closed-form solution is available for the deflection/slope/curvature fields. The procedure presently used is the virtual fields method with “special” virtual fields. It is shown that the unknown parameters are directly extracted with this method since no iterative calculations are required. The parameters are in fact directly equal to the virtual work of the applied loading with the special virtual displacement fields. The headlines of the method are recalled in the first part of the paper. They are then applied in the case of anisotropic bent plates. The accuracy and the stability of the procedure are finally discussed through some relevant examples.     

Digital image correlation and biaxial test on composite material for anisotropic damage law identification

The purpose of the work is to extend the use of non-conventional tests and full field measurements to the identification of an anisotropic damage law. A Digital Image Correlation technique based on a finite element discretization is used to extract planar displacement fields. The reconditioned Equilibrium Gap Method is then used to retrieve a damage law that accounts for shear softening, a specific form suited to the present application. The identification is shown to reduce to a linear system. The example of a biaxial shear test performed on a cruciform specimen is considered. The approach is first qualified by using displacement fields resulting from a non-linear computation with a known damage law. A good agreement is observed between the prescribed and identified laws for distinct parameter settings, even when significant noise is added to the displacement fields. The reconstructed displacement fields coincide perfectly with the measurements. The complete scheme is finally tested considering images taken during an experiment performed on a carbon/carbon composite. The identified damage pattern and the corresponding damage values are similar to post-processed maps using classically identified parameters. The reconstructed displacement field accounts for 95% of the fluctuations observed in the measurements.     

Enriched kinematic fields of cracked structures

Far from the crack tip process zone where non-linear phenomena take place, the mechanical behavior of a cracked medium can be analyzed within the framework of elasticity. Apart from the classical singular stress field associated with the elastic behavior, the effect of a confined process zone is decomposed over a set of (super-singular) fields. Because these fields are indexed by the exponent of their decay with distance from the crack tip, the dominant effect of non-linear mechanisms is characterized by the amplitudes of the first super-singular fields (modes I and II). This approach provides a macroscopic characterization of crack tip non-linearities and describes accurately the displacement field. As an application, the cyclic loading of a cracked elasto-plastic medium is discussed.     

Novel procedure for complete in-plane composite characterization using a single T-shaped specimen

This paper deals with the direct identification of the in-plane elastic properties of orthotropic composite plates from heterogeneous strain fields. The shape of the tested specimen is that of a T subjected to a complex stress state. As a result, the entire set of unknown parameters is directly involved in the strain and displacement responses of the sample. No exact analytical solution is available for such a geometry, and a specific strategy is used to identify the different stiffness components from the whole-field displacements measured over the tested specimen with a suitable optical method. The paper focuses mainly on the experimental aspects of the procedure, and an example of mechanical characterization of a fabric-reinforced composite plate is given.     

Measurement of curved-surface deformation in cylindrical coordinates

Stereo vision is used to measure the strain field of a round tension test specimen in a cylindrical coordinate system. Initially, the displacement fields of the specimen are measured relative to a world coordinate system erected by the stereo vision. Through coordinate transformations, the measured displacement fields expressed in world coordinates are then converted to the displacement fields expressed in cylindrical coordinates. By differentiating the axial and circumferential displacements in the axial and circumferential directions, the axial, circumferential and shear strains are determined. Results indicate that the measured mean value of the axial strains is in good agreement with the measurements of the extensometer and the strain gage. The Poisson's ratio obtained by the circumferential and axial strains is close to .33 in the elastic state. The mean error of the computed shear strain is approximately .03 percent in the smaller elastic deformation and .08 percent in the larger plastic deformation.     

Mixed numerical–experimental technique for orthotropic parameter identification using biaxial tensile tests on cruciform specimens

This paper presents a mixed numerical–experimental method for the identification of the four in-plane orthotropic engineering constants of composite plate materials. A biaxial tensile test is performed on a cruciform test specimen. The heterogeneous displacement field is observed by a CCD camera and measured by a digital image correlation (DIC) technique. The measured displacement field and the subsequently computed strain field are compared with a finite element simulation of the same experiment. The four independent engineering constants are unknown parameters in the finite element model. Starting from an initial value, these parameters are updated till the computed strain field matches the experimental strain field. Two specimen geometries are used: one with a centered hole to increase the strain heterogeneity and one without a hole. It is found that the non-perforated specimen yields the most accurate results.     

Scattering of SH-wave from interface cylindrical elastic inclusion with a semicircular disconnected curve

Scattering of SH wave from an interface cylindrical elastic inclusion with a semicircular disconnected curve is investigated. The solution of dynamic stress concentration factor is given using the Green＇s function and the method of complex variable functions. First, the space is divided into upper and lower parts along the interface. In the lower half space, a suitable Green＇s function for the problem is constructed. It is an essential solution of the displacement field for an elastic half space with a semi-cylindrical hill of cylindrical elastic inclusion while bearing out-plane harmonic line source load at the horizontal surface. Thus, the semicircular disconnected curve can be constructed when the two parts are bonded and continuous on the interface loading the undetermined anti-plane forces on the horizontal surfaces. Also, the expressions of displacement and stress fields are obtained in this situation. Finally, examples and results of dynamic stress concentration factor are given. Influences of the cylindrical inclusion and the difference parameters of the two mediators are discussed.     

Scattering of SH-wave from interface cylindrical elastic inclusion with a semicircular disconnected curve

Scattering of SH wave from an interface cylindrical elastic inclusion with a semicircular disconnected curve is investigated.The solution of dynamic stress concentration factor is given using the Gteen's function and the method of complex variable functions.First,the space is divided into upper and lower parts along the interface.In the lower half space,a suitable Green's function for the problem is constructed.It is an essential solution of the displacement field for an elastic half space with a semi-cylindrical hill of cylindrical elastic inclusion while bearing out-plane harmonic line source load at the horizontal surface.Thus,the semicircular disconnected curve can be constructed when the two parts are bonded and continuous on the interface loading the undetermined anti-plane forces on the horizontal surfaces.Also,the expressions of displacement and stress fields are obtained in this situation.Finally,examples and results of dynamic stress concentration factor are given.Influences of the cylindrical inclusion and the difierence parameters of the two mediators are discussed.     

Learning material law from displacement fields by artificial neural network

The recently developed data-driven approach can establish the material law for nonlinear elastic composite materials(especially newly developed materials) by the generated stress-strain data under different loading paths(Computational Mechanics, 2019). Generally, the displacement(or strain) fields can be obtained relatively easier using digital image correlation(DIC) technique experimentally, but the stress field is hard to be measured. This situation limits the applicability of the proposed data-driven approach. In this paper, a method based on artificial neural network(ANN) to identify stress fields and further obtain the material law of nonlinear elastic materials is presented, which can make the proposed data-driven approach more practical. A numerical example is given to prove the validity of the method. The limitations of the proposed approach are also discussed.     

Three-dimensional elastic displacements induced by a dislocation of polygonal shape in anisotropic elastic crystals

Dislocations and the elastic fields they induce in anisotropic elastic crystals are basic for understanding and modeling the mechanical properties of crystalline solids. Unlike previous solutions that provide the strain and/or stress fields induced by dislocation loops, in this paper, we develop, for the first time, an approach to solve the more fundamental problem—the anisotropic elastic dislocation displacement field. By applying the point-force Green’s function for a three-dimensional anisotropic elastic material, the elastic displacement induced by a dislocation of polygonal shape is derived in terms of a simple line integral. It is shown that the singularities in the integrand of this integral are all removable. The proposed expression is applied to calculate the elastic displacements of dislocations of two different fundamental shapes, i.e. triangular and hexagonal. The results show that the displacement jump across the dislocation loop surface exactly equals the assigned Burgers vector, demonstrating that the proposed approach is accurate. The dislocation-induced displacement contours are also presented, which could be used as benchmarks for future numerical studies.     

A Unified Two-Phase Potential Method for Elastic Bi-material: Planar Interfaces

This paper gives a unified approach to analyze two-dimensional elastic deformations of a composite body consisting of two dissimilar anisotropic or isotropic materials perfectly bonded along a planar interface. The Eshelby et al. formalism of anisotropic elasticity is linked with that of Kolosov-Muskhelishvili for isotropic elasticity by means of two complex matrix functions describing completely the arising elastic fields. These functions, whose elements are holomorphic functions, are defined as the two-phase potentials of the bimaterial. The present work is concerned with bi-materials whose constituent materials occupy the whole space and are connected by a planar interface. The elastic fields arising in such a bimaterial are given by universal relationships in terms of the two-phase potentials. Then, the general results obtained are implemented to study two interesting bimaterial problems: the problem of a uniformly stressed bimaterial with a perfect interfacial bonding, and the interface crack problem of a bimaterial with a general loading. For both problems, all combinations of the elastic properties of the constituent materials are considered. For the first problem, the constraints, which must be imposed between the components of the applied uniform stress fields, are established, so that they are admissible as elastic fields of the bimaterial. For the interface crack problem, the solution is obtained for a general loading applied in the body. Detailed results are given for the case of a remote uniform stress field applied to the bimaterial constituents.     

Applying the Virtual Fields Method to the identification of elasto-plastic constitutive parameters

This paper deals with the identification of constitutive parameters governing elasto-plastic constitutive equations. The procedure used is the virtual fields method. It consists in writing the principle of virtual work with particular virtual fields. A cost function is then deduced. Minimizing this cost function provides the unknown parameters. An important feature is the fact that no finite element analysis is required with this approach. Moreover, it is not sensitive to the distribution of the loading if suitable virtual fields are used. Some numerical examples illustrate the relevancy of the method and its sensitivity to noisy data.     

Cracked elastic layer under compressive mechanical loads

We consider boundary value problem in which an elastic layer containing a finite length crack is under compressive loading. The crack is parallel to the layer surfaces and the contact between crack surfaces are either frictionless or with adhesive friction or Coulomb friction.Based on fourier integral transformation techniques the solution of the formulated problems is reduced to the solution of a singular integral equation, then, using Chebyshev’s orthogonal polynomials, to an infinite system of linear algebraic equations. The regularity of these equations is established. The expressions for stress and displacement components in the elastic layer are presented. Based on the developed analytical algorithm, extensive numerical investigations have been conducted.The results of these investigations are illustrated graphically, exposing some novel qualitative and quantitative knowledge about the stress field in the cracked layer and their dependence on geometric and applied loading parameters. It can be seen from this study that the crack tip stress field has a mode II type singularity.     

The vibration of a half-space due to a buried mode I crack opening

The solution of a dynamic problem for calculation of a displacement field on a half-space surface caused by an internal mode I crack opening is presented. The problem is reduced to the system of boundary integral equations (BIEs). The equations of motion are solved with the use of Helmholtz potentials and applying Fourier integral transform. The effects of the crack size, the crack depth and the distance from the crack epicenter to the observation point on the parameters of elastic waves are investigated. It is established that the increasing of the defect size leads to narrowing bandwidth of elastic waves and to lowering of center frequency. The analysis given here can be used for identification of the crack growth during technical diagnostic of an industry objects and structural elements by AE method.     

THERMAL CONSOLIDATION OF LAYERED POROUS HALF-SPACE TO VARIABLE THERMAL LOADING

An analytical method was derived for the thermal consolidation of layered, saturated porous half-space to variable thermal loading with time. In the coupled governing equations of linear thermoelastic media, the influences of thermo-osmosis effect and thermal filtration effect were introduced. Solutions in Laplace transform space were first obtained and then numerically inverted. The responses of a double-layered porous space subjected to exponential decaying thermal loading were studied. The influences of the differences between the properties of the two layers (e.g., the coefficient of thermal consolidation, elastic modulus) on thermal consolidation were discussed. The studies show that the coupling effects of displacement and stress fields on temperature field can be completely neglected, however, the thermo-osmosis effect has an obvious influence on thermal responses.     

Time domain modeling of damping using anelastic displacement fields and fractional calculus

A fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and elastic part is developed. The evolution equation for the anelastic part is then a differential equation of fractional order in time. By using a fractional order evolution equation for the anelastic strain the present model becomes very flexible for describing the weak frequency dependence of damping characteristics. To illustrate the modeling capability, the model parameters are fit to available frequency domain data for a high damping polymer. By studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning. The use of this viscoelastic model in structural modeling is discussed and the corresponding finite element equations are outlined, including the treatment of boundary conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. Finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented.     

SH波入射半空间双相介质界面附近圆形衬砌的动力分析

利用复变函数和Green函数法研究了无限半空间中双相介质界面附近圆形衬砌对SH波的散射与动应力集中问题。该问题的解答采用镜像法,首先构造出含有圆形衬砌的直角平面区域出平面问题的Green函数,然后利用“契合”技术,并根据界面处位移连续性条件将解答归结为具有弱奇异性的第一类Fredholm积分方程组的求解,结合散射波的衰减特性,直接离散该方程组,把积分方程组转化为线性代数方程组可得到该问题的数值结果。最后,通过算例分析了不同介质参数、几何参数和入射波时圆形衬砌界面的动应力集中情况。     

The radially nonhomogeneous elastic axisymmentric problem

The plane axisymmetric problem with axisymmetric geometry and loading is analyzed for a radially nonhomogeneous circular cylinder, in linear elasticity. Considering the radial dependence of the stress, the displacements fields and of the stiffness matrix, after a series of admissible functional manipulations, the general differential system solving the problem is developed. The isotopic radially inhomogeneous elastic axisymmetric problem is also analyzed. The exact elasticity solution is developed for a radially nonhomogeneous hollow circular cylinder of exponential Young’s modulus and constant Poisson’s ratio and of power law Young’s modulus and constant Poisson’s ratio. For the isotropic elastic axisymmetric problem, a general expression of the stress function is derived. After the satisfaction of the biharmonic equation and making compatible the stress field’s expressions, the stress function and the stress and displacements fields of the axisymmetric problem are also deduced. Applications have been made for a radially nonhomogeneous hollow cylinder where the stress and displacements fields are determined.     

A two-phase model and related numerical tool for the design of soil structures reinforced by stiff linear inclusions

A two-phase macroscopic model is presented for ascertaining the overall linear elastic behaviour of a soil reinforced by stiff long inclusions, with a particular emphasis on the shear and flexural behaviour of the reinforcements. Based on a minimum principle for the potential energy of any two-phase system, expressed as a function of the kinematically admissible displacement and rotation fields, a finite element formulation is established, resulting in the elaboration of a numerical computer code devoted to the simulation of any reinforced soil structure under plane strain conditions. This finite element code is applied to the evaluation of the settlements experienced by a rigid raft foundation placed on top of a soil reinforced by a group of vertical piles, and subject to combined loading conditions. One of the important conclusions which may be drawn from such a quantitative analysis is that the shear and flexural behaviour of the reinforcing piles, as well as the way these piles are connected to the foundation, play a decisive role in the case of lateral loading.