Application of Moiré Interferometry to the Characterization of Orthotropic Materials in the Antisymmetric Configuration using the Virtual Fields Method

Moiré interferometry is an effective full-field deformation measurement technique and has been utilized for mechanical behavior analysis of materials and structures. For isotropic materials, Moiré patterns can be obtained by performing standard tests, such as, tensile and bending tests, to calculate the displacement and strain. Then, the mechanical properties can be characterized. However, standard tests are not sufficient to characterize the mechanical parameters of anisotropic materials due to the complexity of their material properties. Thus, in this work, Moiré interferometry was combined with the Virtual Fields Method to obtain the four in-plane elastic constants (Q₁₁, Q₂₂, Q₁₂, and Q₆₆) of orthotropic materials in the form of a diametrically compressed disk. Firstly, according to finite element method simulation results, optimized parameters can be achieved when the principal direction of the material does not coincide with the loading direction, making the loading configuration antisymmetric. Therefore, Moiré interferometry experiment was simulated to demonstrate the feasibility of measurement in the antisymmetric configuration. Finally, the Q₁₁, Q₂₂, Q₁₂ and Q₆₆ values of a unidirectional carbon fiber composite were measured in a real Moiré interferometry experiment using the proposed method, yielding results that agreed closely with those obtained using the strain gauges.     

Basic equations for the microscopic theory of plasticity of polycrystalline materials with given texture

An expression for the yield stress of anisotropic materials is applied to the anisotropic strength of hard rolled copper foils whose crystallographic texture is known. We assume that this crystallographic texture is the only cause of the anisotropic plastic behaviour of the material. The model used for the yield stress is also used to deduce:

1. Stress-strain relations for isotropic polycrystalline materials;
2. A formula for the fully plastic strain tensor, applied to anisotropic hard rolled copper foils.

For the anisotropic copper foils considered the calculated curves of the yield stress and of the strain tensor as a function of the angle x between rolling and tensile direction agree qualitatively with the measured values. However, the theory is not complete, since the yield stress and the plastic strain tensor are both a function of a parameter Q, the fraction of the number of available crystallographic slip planes on which the maximum shear stress has reached the critical value τ_a. We assume that for “fully” plastic deformation a certain critical fraction Q _e of the total number of slip planes has to be active. The fraction Q _e is called the critical active quantity. With the parameter Q _e we adjust the calculated curves to the measured ones. The dependence of Q _e on the properties of the material (e.g. the crystallographic texture) is discussed in Appendix I.     

Green's function solution and displacement potentials for Transformed Transversely Isotropic materials

A totally non-degenerate expression for the Green's function of infinite Transversely Isotropic (TI) materials is first deduced from the solutions given by Pan and Chou [Pan, Y.-C., Chou, T.-W., 1976. Point force solution for an infinite transversely isotropic solid. Trans. ASME, J. Appl. Mech. 43 (4), 608–612]. Then this solution and also the displacement potentials for TI materials are extended by a linear transformation to a larger family of anisotropic materials (Transformed Transversely Isotropic or TraTI materials). This family depends on 12 independent parameters and contains non-orthotropic materials and in this way a first explicit analytical solution for the Green's function for a non-orthotropic material is obtained. The TraTI materials which have orthotropic Symmetry (StraTI materials) constitute a sub-family depending on 6 independent parameters in the symmetry basis of the material. These materials present a 3D anisotropy (different stiffnesses in three orthogonal directions). General displacement potentials and the Green's function solution for STraTI materials can be deduced by a simple change and introducing one additional parameter in the well-known TI solutions.     

A generating function for the yield criterion of isotropic and anisotropic polycrystalline materials

Summary A yield criterion for elastic pure-plastic polycrystalline materials is generated under simplified conditions by assuming that for yielding a certain fraction Q _c of the total number of slip planes in the material has to be active. This fraction Q _c is called the critical active quantity. We suppose Q _c to be independent of the state of stress. The yield criterion is mathematically expressed as an integral, which is a function of Q _c. This criterion can also be used for anisotropic materials.For isotropic materials the ratio (r) of the yield stress in torsion to that in tension is calculated as a function of Q _c. We find 0.5r0.61.The value r=0.5 (Tresca's criterion) is obtained for Q _c=0 and Q _c=1. The value r=0.577 (von Mises criterion) is obtained for Q _c=0.34 and Q _c=0.79. The difference between two criteria with the same r is the magnitude of the yield stress. We think the value Q _c=0.79 corresponds to the experiments for f.c.c. materials, since a rough estimation gives Q _c>0.75 for yielding.The independence of Q _c on the state of stress brings on that r>0.5 is more probable. This is caused by the slower increase to Q _c in torsion compared with the case of tension.From the theory follows that in the general case (Q _c0) the middle principal stress has influence on yielding.In this paper we don't determine Q _c, but adapt its value to the experimental results. However, a rough estimation of Q _c is given for isotropic materials.     

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p² with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x₂ of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.     

Determination of Anisotropic Plastic Constitutive Parameters Using the Virtual Fields Method

The aim of the present study is to retrieve all the anisotropic plastic constitutive parameters from uniaxial loading. A complex geometry which can provide very heterogeneous stress states in a uniaxial tensile test was chosen for steel sheet specimens. A digital image correlation technique was used for the full-field heterogeneous strain measurement. The orthotropic Hill1948 yield criterion with Swift isotropic hardening was adopted as an elasto-plastic constitutive model. The virtual fields method (VFM) was employed as an inverse analytical tool to determine the constitutive parameters. All the parameters were successfully identified using the VFM by combining two tensile test results obtained in rolling and transverse directions.     

Identification of Material Parameters of PVC Foams using Digital Image Correlation and the Virtual Fields Method

This paper presents an effective methodology to characterize all the constitutive (elastic) parameters of an orthotropic polymeric foam material (Divinycell H100) in one single test using Digital Image Correlation (DIC) in combination with the Virtual Fields Method (VFM). A modified Arcan fixture is used to induce various loading conditions ranging from pure shear or axial loading in tension or compression to bidirectional loading. A numerical optimization study was performed with different loading angles of the Arcan test fixture and off-axis angles of the principal material axes. The objective is to identify the configuration that gives the minimum sensitivity to noise and missing data on the specimen edges, which are the two major issues when identifying the stiffness components from actual DIC measurements. Two optimized Arcan test configurations were chosen. The experimental results obtained for these two optimized test configurations show a significant improvement of the measurement accuracy compared with a pure shear load configuration. The larger sensitivity of the pure shear test to missing data as opposed to the tensile test is also evident from the experimental data and confirms the analysis from the optimization study. The recovery of missing data along the specimen edges is a promising way to further improve the identification results.     

Homogenization of two-phase elasto-plastic composite materials and structures: Study of tangent operators,cyclic plasticity and numerical algorithms

We develop homogenization schemes and numerical algorithms for two-phase elasto-plastic composite materials and structures. A Hill-type incremental formulation enables the simulation of unloading and cyclic loadings. It also allows to handle any rate-independent model for each phase. We study the crucial issue of tangent operators: elasto-plastic (or “continuum”) versus algorithmic (or “consistent”), and anisotropic versus isotropic. We apply two methods of extraction of isotropic tangent moduli. We compare mathematically the stiffnesses of various tangent operators. All rate equations are discretized in time using implicit integration. We implemented two homogenization schemes: Mori–Tanaka and a double inclusion model, and two plasticity models: classical J₂ plasticity and Chaboche’s model with non-linear kinematic and isotropic hardenings. We consider composites with different properties and present several discriminating numerical simulations. In many cases, the results are validated against finite element (FE) or experimental data. We integrated our homogenization code into the FE program ABAQUS using a user material interface UMAT. A two-scale procedure allows to compute realistic structures made of non-linear composite materials within reasonable CPU time and memory usage; examples are shown.     

Limit point instability in pressurization of anisotropic finitely extensible hyperelastic thin-walled tube

Mechanical responses of materials undergoing large elastic deformations can exhibit a loss of stability in several ways. Such a situation can occur when a thin-walled cylinder is inflated by an internal pressure. The loss of stability is manifested by a non-monotonic relationship between the inflating pressure and internal volume of the tube. This is often called limit point instability. The results, known from the literature, show that isotropic hyperelastic materials with limiting chain extensibility property always exhibit a stable response if the extensibility parameter of the Gent model satisfies J_m<18.2. Our study investigates the same phenomenon but for tubes with anisotropic form of the Gent model (finite extensibility of fibers). Anisotropy, used in our study, increases the number of material parameters the consequence of which is to increase degree of freedom of the problem. It will be shown that, in stark contrast to isotropic material, the unstable response is predicted not only for large values of J_m but also for J_m≈1 and smaller, and that the existence of limit point instability significantly depends on the orientation of preferred directions and on the ratio of linear parameters in the strain energy density function (this ratio can be interpreted as the ratio of weights by which fibers and matrix contribute to the strain energy density). Especially tubes reinforced with fibers oriented closely to the longitudinal direction are susceptible to a loss of monotony during pressurization.     

The degeneration of image singularities from anisotropic materials to isotropic materials for an elastic half-plane

The degeneration of image singularities from an anisotropic material to an isotropic material for a half-plane is discussed in this study. The Green’s functions for anisotropic and isotropic half-planes with traction free boundary subjected to concentrated forces and dislocations have been obtained by many authors. It was commonly accepted that the solution of isotropic problem cannot be derived from anisotropic solutions. However, we believe that this possibility exists as we will demonstrate in this paper. Anisotropic materials include only image singularities of order O(1/r) (i.e., forces and dislocations) existing on image points. There are many image points for anisotropic materials and the locations of these image points depend on the material constants. However, isotropic materials have only one image point with higher order image singularities (O(1/r²), O(1/r³)). From the analysis provided in this study, it is found that the higher order image singularities for an isotropic half-plane are generated by combining the concentrated forces and dislocations when an anisotropic material degenerates to an isotropic material. The solutions of higher order image singularities for isotropic material are dependent. Therefore, these image singularities can be combined to form only three or four simpler image singularities acting on an image point of the isotropic material.     

Constitutive behaviour of anisotropic materials under shock loading

Thermodynamically and mathematically consistent constitutive equations suitable for shock wave propagation in an anisotropic material are presented in this paper. Two fundamental tensors α_ij and β_ij which represent anisotropic material properties are defined and can be considered as generalisations of the Kronecker delta symbol, which plays the main role in the theory of isotropic materials. Using two fundamental tensors α_ij and β_ij, the concept of total generalised “pressure” and pressure corresponding to the thermodynamic (equation of state) response are redefined. The equation of state represents mathematical and physical generalisation of the classical Mie–Grüneisen equation of state for isotropic material and reduces to the Mie–Grüneisen equation of state in the limit of isotropy. Based on the generalised decomposition of the stress tensor, the modified equation of state for anisotropic materials, and the modified Hill criteria, combined with the associated flow rule, a system of constitutive equations suitable for shock wave propagation is formulated. The behaviour of aluminium alloy 7010-T6 under shock loading conditions is considered. A comparison of numerical simulations with existing experimental data shows good agreement of the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels, and suggests that the constitutive equations are performing satisfactorily. The results are presented and discussed, and future studies are outlined.     

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

An Eulerian formulation for large deformations of anisotropic elastic and viscoelastic solids and viscous fluids

An Eulerian formulation has been developed for the constitutive response of a group of materials that includes anisotropic elastic and viscoelastic solids and viscous fluids. The material is considered to be a composite of an elastic solid and a viscous fluid. Evolution equations are proposed for a triad of vectors m _i that represent the stretches and orientations of material line elements in the solid component. Evolution equations for an orthonormal triad of vectors s _i are also proposed to characterize anisotropy of the fluid component. In particular, for an elastic solid it is shown that the material response is totally characterized by the functional form of the strain energy and by the current values of m _i , which are measurable in the current state of the material. Moreover, it is shown that the proposed Eulerian formulation removes unphysical arbitrariness of the choice of the reference configuration in the standard formulation of constitutive equations for anisotropic elastic solids.     

An experimental and numerical study on the formability of textured AlZnMg alloys

The influence of texture and grain structure on strain localisation and formability is investigated experimentally and numerically for two AlZnMg alloys. The considered alloys have recrystallised or non-recrystallised grain structure and strong or nearly random texture. The textured materials have rotated cube texture or β-fibre texture of high intensity. A comprehensive test programme, including uniaxial tension tests in three directions, through-thickness compression tests, plane-strain tension tests and double-plate formability tests, is completed to determine the work hardening, plastic anisotropy and formability of the materials. Strain localisation and failure are examined by optical microscopy. Using parts of the test data, an anisotropic plasticity model is calibrated and applied in calculation of forming limit curves, using the Marciniak–Kuczynski (M-K) analysis for anisotropic materials. The formability tests show that the materials with nearly random texture exhibit superior formability. This is mainly attributed to enhanced work hardening for these materials. For the material exhibiting strong β-fibre texture significantly lower formability is found in equibiaxial stretching than in plane strain, while this characteristic is not seen for the material with strong cube texture. The M-K analysis is capable of predicting the major trends of the experiments, and captures the low formability of the alloy with strong β-fibre texture under equibiaxial straining. A numerical study is performed to evaluate the sensitivity of the predicted forming limit curves to parameters not determined experimentally.     

On determination of mode II fracture toughness using semi-circular bend specimen

The cracked semi-circular specimen subjected to three-point bending has been recognized as an appropriate test specimen for conducting mode I, mode II and mixed mode I/II fracture tests in brittle materials. The manufacturing and pre-cracking of the specimen are simple. No complicated loading fixture is also required for a fracture test. However, almost all of the theoretical criteria available for mixed mode brittle fracture fail to predict the experimentally determined mode II fracture toughness obtained from the semi-circular bend (SCB) specimen. In this paper, a modified maximum tangential stress criterion is used for calculating mode II fracture toughness K_IIc in terms of mode I fracture toughness K_Ic. The modified criterion is used for predicting the reported values of mode II fracture toughness for two brittle materials: a rock material (Johnstone) and a brittle polymer (PMMA). It is shown that the modified criterion provides very good predictions for experimental results.     

Non-coaxiality of strain increment and stress directions in cross-anisotropic sand

An experimental program was carried out in a recently developed torsion shear apparatus to study the non-coaxiality of strain increment and stress directions in cross-anisotropic deposits of Fine Nevada sand. Forty-four drained torsion shear tests were performed at constant mean confining stress, σ_m, constant intermediate principal stress ratios, as indicated by b = (σ₂ − σ₃)/(σ₁ − σ₃), and constant principal stress directions, α. The experiments were performed on large hollow cylinder specimens deposited by dry pluviation and tested in an automated torsion shear apparatus. The specimens had height of 40 cm, and average diameter of 20 cm, and wall thickness of 2 cm. The stress–strain behavior of Fine Nevada sand is presented for discrete combinations of constant principal stress direction, α, and intermediate principal stress. The effects of these two variables on the non-coaxiality are presented. The experiments show that the directions of the strain increments do not in general coincide with the directions of stresses, and there is a switch from one to the other side between the two quantities.     

Surface responses induced by point load or uniform traction moving steadily on an anisotropic half-plane

Surface responses induced by point load or uniform traction moving steadily with subsonic speed on an anisotropic half-plane boundary are investigated. It is found that the effects of the material constant on surface displacements are through matrices L^?1(v) and S(v)L^?1(v), while those on surface stress components are through matrices Ω(v) and Γ(v). Explicit expressions for the elements of these four matrices are expressed in terms of elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with symmetry plane at x₁ = 0, x₂ = 0 and x₃ = 0, and the case for orthotropic materials are all deduced. Results for isotropic material may be recovered from present results. For monoclinic materials with a plane of symmetry at x₃ = 0, two of the elements of matrix Ω(v) are found to be independent of subsonic speed.     

On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

Ellipsoidal anisotropy in linear elasticity: Approximation models and analytical solutions

The concept of ellipsoidal anisotropy, first introduced in linear elasticity by Saint Venant, has reappeared in recent years in diverse applications from the phenomenological to micromechanical modeling of materials. In this concept, indicator surfaces, which represent the variation of some elastic parameters in different directions of the material, are ellipsoidal. This concept recovers different models according to the elastic parameters that have ellipsoidal indicator surfaces. An interesting feature of some models introduced by Saint Venant is the formation of analytical solutions for basic problems of linear elasticity. This paper has two main objectives. First, an accurate definition of the variety of anisotropy called ellipsoidal is provided, which corresponds to a family of materials that depends on 12 independent parameters, including varieties of orthotropic and non-orthotropic materials. An explicit nondegenerate Green function solution is established for these materials. Then, it is shown that the ellipsoidal model recovers a variety of phenomenological and theoretical models used in recent years, specifically for geomaterials and damaged or micro-cracked materials. These models can be used to approximate the elastic parameters of any anisotropic material with different fitting qualities. A method to optimize the parameters will be given.     

Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials

Novel (non-quadratic) plasticity criteria based on Kelvin modes are formulated here for anisotropic materials. As an example, such a macroscopic criterion is applied with success to the case of FCC nickel-base single crystals. Indeed, relying on the cubic symmetry of the material, the Kelvin decomposition of elasticity tensor easily allows for the definition of an objective and loading independent criterion. The criterion identification is performed from different loading cases for CMSX2 single crystal superalloy. Tension-torsion yield surfaces at room temperature and yield stress dependence on crystal orientation are modeled. The Kelvin modes based criterion is compared to experimental data, to Hill and Barlat and coworkers macroscopic criteria and to Schmid law predictions. The results show that a simple three-parameter yield function built thanks to von Mises equivalent Kelvin stresses accounts for a satisfying plasticity criterion for such alloys.Non-quadratic norm ∥·∥_a plasticity framework is addressed. Intrinsic generalizations of Hershey-Hosford criterion are proposed for cubic material symmetry.