Effect of nonlinear response of concrete on its elastic modulus and strength

The purpose of this paper is to investigate the effect of nonlinear response of concrete on the relationship between modulus of elasticity at static and dynamic loading as well as on its strength. The obtained relationships are based on the thermofluctuation strength theory coupled with a nonlinear stress–strain material model. From the corresponding equations it was found that the ratio of the static to the dynamic modulus of elasticity depends on the strength of concrete, its temperature, rate of loading. Also it was confirmed that the dynamic modulus is greater than static modulus of elasticity. These equations explain the influence of the value of applied stress on the value of the static or dynamic modulus of elasticity. Comparative study shows substantial agreement with existing experimental results and the general equations given in standard BS 8110, Part 2:1985, and ACI documents. Based on the obtained relationships new methods for evaluating the static modulus of concrete and its strength from the results of dynamic tests are described subsequently.     

Transient analysis of the DSIFs and dynamic T-stress for particulate composite materials—Numerical vs. experimental results

The fracture behavior of particulate composite materials when subjected to dynamic loading has been a great concern for many industrial applications as these materials are particularly susceptible to impact loading conditions. As a result, many numerical and experimental techniques have been developed to deal with this class of problems. In this work, the fracture behavior of particulate composites under impact loading conditions is numerically studied via the two most important fracture parameters: dynamic stress intensity factors (DSIFs) and dynamic T-stress (DTS), and the results are compared with the experimental data obtained in Refs. [1,2]. Here, micromechanics models (self-consistent, Mori–Tanaka, …) or experimental techniques need to be employed first to determine the effective material properties of particulate composites. Then, the symmetric-Galerkin boundary element method for elastodynamics in the Fourier-space frequency domain is used in conjunction with displacement correlation technique to evaluate the DSIFs and stress correlation technique to determine the DTS. To obtain transient responses of the fracture parameters, fast Fourier transform (FFT) and inverse FFT are subsequently used to convert the DSIFs and DTS from the frequency domain to the time domain. Test examples involving free–free beams made of particulate composites are considered in this study. The numerical results are found to agree very well with the experimental ones.     

Investigation of singularity orders and eigen-angular functions for V-notches in radially inhomogeneous materials

The singularities of V-notches in a material whose modulus of elasticity varies as a power function of the r-coordinate are investigated. The eigen-equations are derived by substituting the asymptotic expansions of displacement fields around a notch vertex into the elasticity equilibrium equations. The radial boundary conditions are also presented by a combination of stress singularity orders and eigen-angular functions. The singularity orders and eigen-angular functions can be calculated simultaneously by solving a set of eigen ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method developed by some of the authors before without the iterative process. The accuracy of the proposed method is verified by comparing the present results with the reference ones when the inhomogeneous material is degraded into a homogeneous one. Then, the stress singularities of V-notches in the radially inhomogeneous material under the plane and anti-plane loadings are investigated, respectively. The results show that the stress singularity of a V-notch in the radially inhomogeneous material under the plane loading is more serious than the one under the anti-plane loading. The plane V-notch under the clamped–clamped boundary condition presents the stress singularity at a smaller notch angle α than the one under the free–free boundary condition. The radially inhomogeneous bulgy V-notch even presents singularity. In addition, the stress singularity becomes stronger with the increase of the exponent c in the variation function of the elasticity modulus.     

T-shaped crack problem for bonded orthotropic layers

The plane elasticity problem of two perfectly bonded orthotropic layers containing cracks perpendicular to and along the interface is considered. Cracks are extended to intersect the boundaries and each other in such a way that a crack configuration suitable to study the T-shaped crack problem is obtained. The problem is reduced to the solution of a system of singular integral equations with Cauchy type singularities. Numerical results for stress intensity factors and energy release rates are presented for various loading conditions and for isotropic as well as orthotropic material pairs. These results indicate that elementary strength of material type calculations for energy release rates provide a good approximation to the actual elasticity solution even for relatively short cracks, as long as the layer thicknesses are not very different.     

Micromechanical study of the loading path effect in high cycle fatigue

In this work, an analysis of both the mechanical response at the grain scale and high cycle multiaxial fatigue criteria is undertaken using finite element (FE) simulations of polycrystalline aggregates. The metallic material chosen for investigation, a pure copper, has a Face Centred Cubic (FCC) crystalline structure. Two-dimensional polycrystalline aggregates, which are composed of 300 randomly orientated equiaxed grains, are loaded at the median fatigue strength defined at 10⁷ cycles. In order to analyse the effect of the loading path on the local mechanical response, combined tension–torsion and biaxial tension loading cases, in-phase and out-of-phase, with different biaxiality ratios, are applied to each polycrystalline aggregate. Three different material constitutive models assigned to the grains are investigated: isotropic elasticity, cubic elasticity and crystal plasticity in addition to the cubic elasticity. First, some aspects of the mechanical response of the grains are highlighted, namely the scatter and the multiaxiality of the mesoscopic responses with respect to an uniaxial macroscopic response. Then, the distributions of relevant mechanical quantities classically used in fatigue criteria are analysed for some loading cases and the role of each source of anisotropy on the mechanical response is evaluated and compared to the isotropic elastic case. In particular, the significant influence of the elastic anisotropy on the mesoscopic mechanical response is highlighted. Finally, an analysis of three different fatigue criteria is conducted, using mechanical quantities computed at the grain scale. More precisely, the predictions provided by these criteria, for each constitutive model studied, are compared with the experimental trends observed in metallic materials for such loading conditions.     

AN ANALYTICAL STUDY OF VOID GROWTH IN VISCOPLASTIC SOLIDS

Detailed numerical analyses are conducted to examine the effect of material elasticity and non-linear shape changes on void growth in viscoplastic solids. Numerical simulations are carried out under constant stress as well as cyclic loading conditions. The material response is assumed to be elastic-non-linear viscous (power law creep). Analyses of void growth are conducted assuming creep-controlled mechanism. The role of finite geometry changes in understanding cavity growth under balanced strain-controlled cyclic loading is also explored. The results indicate that while elastic accommodation does not influence void growth, non-linear shape changes of the voids significantly affect their growth characteristics. It is also shown that while grain boundary voids can grow under balanced strain cyclic loading when the accompanying shape changes are accounted for, they can either shrink or grow under load-controlled cycling depending on the loading frequency.     

Steady-state propagation of a mode II crack in couple stress elasticity

The present work deals with the problem of a semi-infinite crack steadily propagating in an elastic body subject to plane-strain shear loading. It is assumed that the mechanical response of the body is governed by the theory of couple-stress elasticity including also micro-rotational inertial effects. This theory introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying microstructure and has proved to be very effective for modeling complex microstructured materials. It is assumed that the crack propagates at a constant sub-Rayleigh speed. An exact full field solution is then obtained based on integral transforms and the Wiener–Hopf technique. Numerical results are presented illustrating the dependence of the stress intensity factor and the energy release rate upon the propagation velocity and the characteristic material lengths in couple-stress elasticity. The present analysis confirms and extends previous results within the context of couple-stress elasticity concerning stationary cracks by including inertial and micro-inertial effects.     

A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials under thermal shock

In the present work, a study of thermoelastic analysis of functionally graded cylindrical shells subjected to transient thermal shock loading is carried out. A semi-analytical axisymmetric finite element model using the three-dimensional linear elasticity theory is developed. The three-dimensional equations of motion are reduced to two-dimensional ones by expanding the displacement field in Fourier series in the circumferential direction involving circumferential harmonics. The material properties are graded in the thickness direction according to a power law. The model has been verified with the results of simple analytical isotropic cylindrical shells subjected to a transient thermal loading. Additional FGM results for stresses and displacements are presented.     

Switching between Elasticity and Plasticity by Network Strength Competition

Switching a material between highly elastic and plastic would be of great use in many fields but has proven to be extremely challenging. Here, the use of mechanical strength competition between two networks in a hybrid material is reported to switch between elasticity and plasticity. In a gel material composed of an elastic polymer network and a shear-thinning nanofiber network, the excellent elasticity of the gel is demonstrated when the former is stronger than the latter. In contrast, the gel exhibits an extraordinary plasticity, which can be stretched to form a permanent anisotropic and tough gel due to the orientation of the nanofibers. The mechanical strength of each network can be simply tuned by adjusting either the crosslinking density or the loading of the nanofibers. This work may open a window to transform a material between superior elastic and plastic, which is useful for the development of adaptable materials.     

Thermoelastic and infrared-thermography methods for surface strains in cracked orthotropic composite materials

Two quantitative thermoelastic strain analysis (TSA) experimental methods are proposed to determine the surface strain fields in mechanically loaded orthotropic materials using the spatial distribution of temperature gradient measured from the surface. Cyclic loadings are applied to orthotropic composite specimens to achieve adiabatic conditions. The small change in surface temperatures that resulted from the change in the elastic strain energy is measured using a high sensitivity infrared (IR) camera that is synchronized with the applied loading. The first method is applied for layered orthotropic composites with a coat layer made of isotropic or in-plane transversely isotropic material. In this case, one material parameter (pre-calibrated from the surface) is required to map the strain invariant to the temperature gradients. The proposed method can be used together with Lekhnitskii’s elasticity solution to quantify the full strain field and determine mixed-mode stress intensity factors (SIFs) for crack tips in composite plates subjected to off-axis loading. The second method is formulated for orthotropic layers without a coat and it requires thermo-mechanical calibrations for two material parameters aligned with the material axes. The virtual crack closure technique (VCCT), Lekhnitskii’s and Savin’s elasticity solutions, and finite element (FE) analyses are used for demonstrations and validations of the second experimental method. The SIFs from the TSA methods are very sensitive to the uncertainty in the location of the crack tip and the unknown inelastic or damage zone size around the crack tip. The two experimental methods are effective in generating the strain fields around notched and other FRP composites.     

Time-dependent aspects of the mechanical properties of plant and vegetative tissues

Based on a single regular cell structural model, the effects of loading rate on the compressive behaviour of plant and vegetative tissues have been qualitatively investigated. The cell walls were treated as a polymeric composite material with microfibrils embedded in the highly structured cell wall matrix. The rubber elasticity, the turgor permeability and the loading rate were taken into account to qualitatively predict the tissue stiffness, cell wall stress, turgor pressure, cell debonding force, and the percentage of weight loss of the cell fluid. The predicted results are consistant with the related experimental phenomena.     

An approximating model of failure of structural materials and a problem of a failure wave

A method of numerical calculation of a failure wave in a brittle material in pulsed loading is proposed. The method is based on the variation of the elasticity modulus in transition through the wave front. The results of calculating the movement of a spherical failure wave are presented.Translated from Problemy Prochnosti, No. 9, pp. 4–6, September, 1991.     

A Periodic Array of Cracks in a Functionally Graded Nonhomogeneous Medium Loaded under in-Plane Normal and Shear

In this paper, the problem of a periodic array of parallel cracks in a functionally graded medium is investigated based on the theory of plane elasticity for a nonhomogeneous continuum. Both the in-plane normal (mode I) and shear (mode II) loading conditions are considered. It is assumed that the material nonhomogeneity is represented as the spatial variation of the shear modulus in the form of an exponential function along the direction of cracks, and the Poisson's ratio is constant. For each of the individual loading modes, a hypersingular integral equation is derived, in a separate but parallel manner in which the crack surface displacements are the unknown functions. As the basic parameters in applying the linear elastic fracture mechanics criteria, the mode I and mode II stress intensity factors are defined from the stress fields with the square-root singularity ahead of the crack tips. Numerical results are obtained to illustrate the variations of the stress intensity factors as a function of the crack periodicity for different values of the material nonhomogeneity. The crack surface displacements are also presented for the prescribed loading, material, and geometric combinations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Crack propagation in a strip of material under plane extension

The problem of a uniformly propagating finite crack in a strip of elastic material is solved using the dynamic equations of elasticity in two-dimensions. Two specific conditions of loading on the strip with finite width are discussed. In the first case, the rigidly clamped edges are pulled apart in the opposite directions. The second case considers equal and opposite tractions applied to the crack surface. By varying the strip width to the crack length ratio, the amplitude of the dynamic stresses ahead of the running crack is determined as a function of the crack velocity. The local dynamic stresses are found to be lower than the corresponding static values for the displacement loading condition and higher for the stress loading condition. This effect becomes increasingly more important as the crack length to strip width ratio is enlarged. Numerical results for the dynamic crack opening displacement are also presented.     

Effect of variations in the plate modulus of elasticity on the failure modes of FRP plated R.C. beams

In the absence of FRP plate/glue/concrete interface bond failure (i.e. interfacial debonding), eight possible flexural modes of failure are identified for reinforced concrete beams experiencing lateral loading, and strengthened in flexure with external FRP or steel plates glued to their soffits. All possible changes in such modes of failure, as a result of variations in the modulus of elasticity of the FRP material (assuming an associated constant value of ultimate tensile strength for the FRP plate in a given beam design), have been addressed in some detail, with a quantitative treatment of the critical values of the FRP modulus of elasticity associated with various failure mode transitions (i.e. changes).     

Effective in situ material properties of micron-sized SiO2 particles in SiO2 particulate polymer composites

Several analytical models exist for determination of the Young’s modulus and coefficient of thermal expansion (CTE) of particulate composites. However, it is necessary to provide accurate material properties of the particles as input data to such analytical models in order to precisely predict the composite’s properties, particularly at high particle loading fractions. In fact, the constituent’s size scale often presents a technical challenge to accurately measure the particles’ properties such as Young’s modulus or CTE. Moreover, the in situ material properties of particles may not be the same as the corresponding bulk properties when the particles are embedded in a polymer matrix. To have a better understanding of the material properties and provide useful insight and design guidelines for particulate composites, the concept of “effective in situ constituent properties” and an indirect method were employed in this study. This approach allows for the indirect determination of the particle’s in situ material properties by combining the experimentally determined composite and matrix properties and finite element (FE) models for predicting the corresponding composite properties, then backing out the effective in situ particle properties. The proposed approach was demonstrated with micron-size SiO₂ particle reinforced epoxy composites over a range of particle loading fractions up to 35 vol.% by indirectly determining both the effective Young’s modulus and the effective CTE of the particles. To the best of our knowledge, this study is the first published report on the indirect determination of both the Young’s modulus and the CTE of micron size particles in particulate composites. Similar results on Young’s modulus of micron-size SiO₂ particles measured from nano-indentation testing are encouraging.     

Generalisation of unidirectional loop layout problem and solution by a genetic algorithm

Unidirectional loop layouts (ULLs) are the preferred layouts in manufacturing systems owing to their relative low investment costs, high material handling elasticity and routing flexibility. Existing formulations of the unidirectional loop layout problem are concentrated on the arrangement of workstations under the assumption that the number and location of loading and unloading stations are known. In this study, the unidirectional loop layout problem is generalised by consideration of potentially attachable loading/unloading equipment to each workstation and releasing of the predetermined number of loading and unloading stations. Thus, more efficient and effective loop layout designs are allowed by eliminating some artificial restrictions. The present ULL model is generalised and a genetic algorithm is developed to solve the problem. Solutions obtained by the genetic algorithm outperformed those obtained by conventional methods. Additionally, comparisons of the generalised model with existing models on randomly generated test problems yielded encouraging results.     

Buckling of a functionally graded coating with an embedded crack bonded to a homogeneous substrate

In an attempt to simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a functionally graded coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an isotropic stress-strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to give the critical buckling strain and the corresponding crack opening displacement shapes. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions and coating thicknesses.     

Nonlocally related PDE systems for one-dimensional nonlinear elastodynamics

Complete dynamical PDE systems of one-dimensional nonlinear elasticity satisfying the principle of material frame indifference are derived in Eulerian and Lagrangian formulations. These systems are considered within the framework of equivalent nonlocally related PDE systems. Consequently, a direct relation between the Euler and Lagrange systems is obtained. Moreover, other equivalent PDE systems nonlocally related to both of these familiar systems are obtained. Point symmetries of three of these nonlocally related PDE systems of nonlinear elasticity are classified with respect to constitutive and loading functions. Consequently, new symmetries are computed that are: nonlocal for the Euler system and local for the Lagrange system; local for the Euler system and nonlocal for the Lagrange system; nonlocal for both the Euler and Lagrange systems. For realistic constitutive functions and boundary conditions, new dynamical solutions are constructed for the Euler system that only arise as symmetry reductions from invariance under nonlocal symmetries.