Showing papers on "Bending moment published in 2000"

Strength of concrete filled steel box columns incorporating local buckling

Abstract: Concrete filled steel box columns have recently experienced a renaissance in their use throughout the world. This has occurred due to the significant advantages that the construction method can provide. This paper deals with the strength behavior of short columns under the combined actions of axial compression and bending moment. The paper addresses the effect of steel plate slenderness limits on this behavior. An extensive set of experiments has been carried out and a numerical model developed elsewhere is augmented and calibrated with these results. A simple model for the determination of the strength-interaction diagram is also verified against both the test results and the numerical model developed in this paper. This model, based on the rigid plastic method of analysis, is existent in international codes of practice, but does not account for the effects of local buckling, which are found to be significant with large plate slenderness values, particularly for large values of axial force. Thus some suggested modifications are proposed to allow for the inclusion of slender plated columns in design.

Biomechanics of the Mandible

Abstract: In this review the biomechanical behavior of the mandibular bone tissue, and of the mandibular bone as a whole, in response to external loading is discussed. A survey is given of the determinants of mandibular stiffness and strength, including the mechanical properties and distribution of bone tissue and the size and shape of the mandible. Mandibular deformations, stresses, and strains that occur during static biting and chewing are reviewed. During biting and the powerstroke of mastication, a combination of sagittal bending, corpus rotation, and transverse bending occurs. The result is a complex pattern of stresses and strains (compressive, tensile, shear, torsional) in the mandible. To be able to resist forces and bending and torsional moments, not only the material properties of the mandible but also its geometrical design is of importance. This is reflected by variables like polar and maximum and minimum moments of inertia and the relative amount and distribution of bone tissue. In the longitudinal direction, the mandible is stiffer than in transverse directions, and the vertical cross-sectional dimension of the mandible is larger than its transverse dimension. These features enhance the resistance of the mandible to the relatively large vertical shear forces and bending moments that come into play in the sagittal plane.

Behavior of High-Strength Concrete Columns under Cyclic Flexure and Constant Axial Load

Abstract: In this study, six large-scale columns made of high-strength concrete (HSC) were subjected to constant axial loads corresponding to target values of 15%, 25%, and 40% of the column axial-load capacity and a cyclic horizontal load-inducing reversed bending moment. Results reveal that tie spacing, and therefore tie volumetric ratio and axial-load level, have significant effects on the flexural behavior of HSC columns. The need to include the axial-load level in code requirements for confinement reinforcement is pointed out.

Performance of fiber-reinforced polymer-wrapped reinforced concrete column under combined axial-flexural loading

Abstract: Many countries around the world have the tremendous need to repair and strengthen their existing infrastructure. This paper presents results of an experimental investigation into the performance of reinforced concrete beam-columns strengthened with externally applied bidirectional carbon fiber-reinforced polymer (CFRP) material. The external moment was applied to the specimens through corbels that were part of the columns. The overall length of the column specimens, including the corbels, was 11.8 ft (3.6 m). Six series of tests were performed on the specimens. The first five series, corresponding respectively to eccentricities of 0, 3, 6, 12, and 16 in. (0, 75, 150, 300, and 400 mm), were performed under a combined axial-flexural loading condition. The sixth series was tested in four-point pure flexure with no axial load. Results indicate that the strength capacity of beam-columns improved significantly as a result of the combined action of the longitudinal and the transverse weaves of the bidirectional composite fabric. The longitudinal CFRP elements contributed mostly to flexural capacity, whereas the transverse elements enhanced the compressive capacity of the compression zone through confinement action. The maximum capacity gain achieved was slightly below 30% in pure compression, and over 54% in pure flexure. Under combined axial force-bending moment conditions, the gain in moment capacity attained 70%. The increase in the compressive strain attributed to the confinement effect varied from 49% to 166%. The transverse confinement was engaged in the compression zone from the early stage of loading. Finally, within the conditions and the limits of this study, the proposed design procedure, based on the stress of confined concrete in the compression zone in conjunction with an effective confinement ratio that takes into account the rectangular shape of the beam-columns, compared reasonably well with experimental results.

Behavior of Pile Subject to Excavation-Induced Soil Movement

Abstract: This paper presents the results of centrifuge model tests on unstrutted deep excavation in dense sand and its influence on an adjacent single pile foundation behind the retaining wall. It is found that, in the case of a stable wall, the induced pile bending moment and deflection decrease exponentially with increasing distance between the pile and the wall. Pile head boundary condition plays an important role in affecting the pile responses due to an adjacent excavation. In the case of retaining wall collapse, the failure pattern of the soil behind the wall features a slip plane projecting from near the wall toe to the ground surface. Soil within the failure zone demonstrates large lateral movement and induces significant bending moment and deflection on pile located within the zone. Soil movement and pile responses outside this zone are noted to be significantly less. A comparison between the experimental results and the theoretical predictions by an existing numerical method shows good agreement, provided that appropriate assumptions are made on the soil parameters and conditions, especially in the case of retaining wall collapse.

Effect of plate position relative to bending direction on the rigidity of a plate osteosynthesis. A theoretical analysis.

Abstract: Mechanical unloading of the plated bone segment is observed after plate osteosynthesis because the implant takes over a part of the physiological loading. Strain reduction in the bony tissue depends on the rigidity of the plate (cross-sectional area, geometrical form, and modulus of elasticity). The aim of the present study was to calculate theoretically the effect of plate position relative to bending direction on the overall bending stiffness of the composite system plate-bone. To calculate the rigidity, a cylindrical bone model with mechanical characteristics similar to a sheep tibia and a rectangular plate cross-section corresponding to a DC-plate with either a modulus of elasticity of steel or titanium was used. Calculations under different bending directions were performed according to the laws of the linear bending theory and the composite beam theory. The bending stiffness of a plate osteosynthesis reaches a minimum and a maximum respectively, in cases in which the bending moment acts in the direction of the main axis of the area moment of inertia of the plate. The minimum is present with the plate bent vertically, the maximum with the plate bent horizontally, e.g. on the tension side of the composite system--on the assumption that the bone structure opposite the plate is capable of withstanding compressive loading. For steel and titanium plates, factors of 2 and 2.25 respectively were calculated between the minimum and the maximum bending stiffnesses of the osteosynthesis. The bending rigidity of the plate alone has only a minimal effect on the total stiffness of the osteosynthesis. With a plate bent vertically, the difference between steel and titanium plates was 18%, with the plate bent horizontally (situated on the tension side), it was only 7%. The bending stiffness of a plate osteosynthesis depends on the cross-section, the geometrical form, and the modulus of elasticity of the plate, as well as on the plate position relative to the bending direction of the composite system. The modulus of elasticity of the plate is relatively unimportant, while with a given plate the individual plate position relative to the bending direction is of crucial importance. Thus, changing the modulus of elasticity of the plate cannot solve the problem of implant induced unloading of the bone cortex because the bending stiffness of the composite system depends much more on the plate position relative to the bending direction.

Optimal placement of PZT actuators for the control of beam dynamics

Abstract: The dynamics of a flexural beam actuated by induced strain surface bonded (piezoelectric) actuators is considered. The bending moment produced by the single actuator is evaluated by means of the pin-force model. A modal approach is then used to build special dynamic influence functions which explicitly account for the size and the position of the actuator. Simple optimal geometrical conditions are then obtained and illustrated for several cases with different boundary conditions.

Intermediate Mechanics of Materials

Abstract: Introduction.- 1.1 The Engineering design process 1.2 Design optimization 1.2.1 Predicting the behaviour of the component.- 1.2.2 Approximate solutions.- 1.3 Relative magnitude of different effects.- 1.4 Formulating and solving problems.- 1.4.1 Use of procedures.- 1.4.2 Inverse problems.- 1.4.3 Physical uniqueness and existence arguments.- 1.5 Review of elementary mechanics of materials.- 1.5.1 Definition of stress components.- 1.5.2 Transformation of stress components.- 1.5.3 Displacement and strain.- 1.5.4 Hooke's law.- 1.5.5 Bending of beams.- 1.5.6 Torsion of circular bars.- 1.6 Summary.- Problems.- 2 Material Behaviour and Failure.- 2.1 Transformation of stresses.- 2.1.1 Review of two-dimensional results.- 2.1.2 Principal stresses in three dimensions.- 2.2 Failure theories for isotropic materials.- 2.2.1 The failure surface.- 2.2.2 The shape of the failure envelope.- 2.2.3 Ductile failure (yielding).- 2.2.4 Brittle failure.- 2.3 Cyclic loading and fatigue.- 2.3.1 Experimental data.- 2.3.2 Statistics and the size effect.- 2.3.3 Factors influencing the design stress.- 2.3.4 Effect of combined stresses.- 2.3.5 Effect of a superposed mean stress.- 2.3.6 Summary of the design process.- 2.4 Summary.- Problems.- 3 Energy Methods.- 3.1 Work done on loading and unloading.- 3.2 Strain energy.- 3.3 Load-displacement relations.- 3.3.1 Beams with continuously varying bending moments.- 3.3.2 Axial loading and torsion.- 3.3.3 Combined loading.- 3.3.4 More general expressions for strain energy.- 3.3.5 Strain energy associated with shear forces in beams.- 3.4 Potential energy.- 3.5 The principle of stationary potential energy.- 3.5.1 Potential energy due to an external force.- 3.5.2 Problems with several degrees of freedom.- 3.5.3 Non-linear problems.- 3.6 The Rayleigh-Ritz method.- 3.6.1 Improving the accuracy.- 3.6.2 Improving the back of the envelope approximation.- 3.7 Castigliano's first theorem.- 3.8 Linear elastic systems.- 3.8.1 Strain energy.- 3.8.2 Bounds on the coefficients.- 3.8.3 Use of the reciprocal theorem.- 3.9 The stiffness matrix.- 3.9.1 Structures consisting of beams.- 3.9.2 Assembly of the stiffness matrix.- 3.10 Castigliano's second theorem.- 3.10.1 Use of the theorem.- 3.10.2 Dummy loads.- 3.10.3 Unit load method.- 3.10.4 Formal procedure for using Castigliano's second theorem.- 3.10.5 Statically indeterminate problems.- 3.10.6 Three-dimensional problems.- 3.11 Summary.- Problems.- 4 Unsymmetrical Bending.- 4.1 Stress distribution in bending.- 4.1.1 Bending about the x-axis only.- 4.1.2 Bending about the y-axis only.- 4.1.3 Generalized bending.- 4.1.4 Force resultants.- 4.1.5 Uncoupled problems.- 4.1.6 Coupled problems.- 4.2 Displacements of the beam.- 4.3 Second moments of area.- 4.3.1 Finding the centroid.- 4.3.2 The parallel axis theorem.- 4.3.3 Thin-walled sections.- 4.4 Further properties of second moments.- 4.4.1 Coordinate transformation.- 4.4.2 Mohr's circle of second moments.- 4.4.3 Solution of unsymmetrical bending problems in principal coordinates.- 4.4.4 Design estimates for the behaviour of unsymmetrical sections.- 4.4.5 Errors due to misalignment.- 4.5 Summary.- Problems.- 5 Non-linear and Elastic-Plastic Bending.- 5.1 Kinematics of bending.- 5.2 Elastic-plastic constitutive behaviour.- 5.2.1 Unloading and reloading.- 5.2.2 Yield during reversed loading.- 5.2.3 Elastic-perfectly plastic material.- 5.3 Stress fields in non-linear and inelastic bending.- 5.3.1 Force and moment resultants.- 5.4 Pure bending about an axis of symmetry.- 5.4.1 Symmetric problems for elastic-perfectly plastic materials.- 5.4.2 Fully plastic moment and shape factor.- 5.5 Bending of a symmetric section about an orthogonal axis.- 5.5.1 The fully plastic case.- 5.5.2 Non-zero axial force.- 5.5.3 The partially plastic solution.- 5.6 Unsymmetrical plastic bending.- 5.7 Unloading, springback and residual stress.- 5.7.1 Springback and residual curvature.- 5.7.2 Reloading and shakedown.- 5.8 Limit analysis in the design of beams.- 5.8.1 Plastic hinges.- 5.8.2 Indeterminate problems.- 5.9 Summary.- Problems.- 6 Shear and Torsion of Thin-walled Beams.- 6.1 Derivation of the shear stress formula.- 6.1.1 Choice of cut and direction of the shear stress.- 6.1.2 Location and magnitude of the maximum shear stress.- 6.1.3 Welds, rivets and bolts.- 6.1.4 Curved sections.- 6.2 Shear centre.- 6.2.1 Finding the shear centre.- 6.3 Unsymmetrical sections.- 6.3.1 Shear stress for an unsymmetrical section.- 6.3.2 Determining the shear centre.- 6.4 Closed sections.- 6.4.1 Determination of the shear stress distribution.- 6.5 Pure torsion of closed thin-walled sections.- 6.5.1 Torsional stiffness.- 6.5.2 Design considerations in torsion.- 6.6 Finding the shear centre for a closed section.- 6.6.1 Twist due to a shear force.- 6.6.2 Multicell sections.- 6.7 Torsion of thin-walled open sections.- 6.7.1 Loading of an open section away from its shear centre.- 6.8 Summary.- Problems.- 7 Beams on Elastic Foundations 7.1 The governing equation.- 7.1.1 Solution of the governing equation.- 7.2 The homogeneous solution.- 7.2.1 The semi-infinite beam.- 7.3 Localized nature of the solution.- 7.4 Concentrated force on an infinite beam.- 7.4.1 More general loading of the infinite beam.- 7.5 The particular solution.- 7.5.1 Uniform loading.- 7.5.2 Discontinuous loads.- 7.6 Finite beams.- 7.7 Short beams.- 7.8 Summary.- Problems.- 8 Membrane Stresses in Axisymmetric Shells.- 8.1 The meridional stress.- 8.1.1 Choice of cut.- 8.2 The circumferential stress.- 8.2.1 The radii of curvature.- 8.2.2 Sign conventions.- 8.3 Self-weight.- 8.4 Relative magnitudes of different loads.- 8.5 Strains and Displacements.- 8.5.1 Discontinuities.- 8.6 Summary.- Problems.- 9 Axisymmetric Bending of Cylindrical Shells.- 9.1 Bending stresses and moments.- 9.2 Deformation of the shell.- 9.3 Equilibrium of the shell element.- 9.4 The governing equation.- 9.4.1 Solution strategy.- 9.5 Localized loading of the shell.- 9.6 Shell transition regions.- 9.6.1 The cylinder/cone transition.- 9.6.2 Reinforcing rings.- 9.7 Thermal stresses.- 9.8 The ASME pressure vessel code.- 9.9 Summary.- Problems.- 10 Thick-walled Cylinders and Disks.- 10.1 Solution method.- 10.1.1 Stress components and the equilibrium condition.- 10.1.2 Strain, displacement and compatibility.- 10.1.3 The elastic constitutive law.- 10.2 The thin circular disk.- 10.3 Cylindrical pressure vessels.- 10.4 Composite cylinders, limits and fits.- 10.4.1 Solution procedure.- 10.4.2 Limits and fits.- 10.5 Plastic deformation of disks and cylinders.- 10.5.1 First yield.- 10.5.2 The fully-plastic solution.- 10.5.3 Elastic-plastic problems.- 10.5.4 Other failure modes.- 10.5.5 Unloading and residual stresses.- 10.6 Summary.- Problems.- 11 Curved Beams.- 11.1 The governing equation.- 11.1.1 Rectangular and circular cross sections.- 11.1.2 The bending moment.- 11.1.3 Composite cross sections.- 11.1.4 Axial loading.- 11.2 Radial stresses.- 11.3 Distortion of the cross section.- 11.4 Range of application of the theory.- 11.5 Summary.- Problems.- 12 Elastic Stability.- 12.1 Uniform beam in compression.- 12.2 Effect of initial perturbations.- 12.2.1 Eigenfunction expansions.- 12.3 Effect of lateral load (beam-columns).- 12.4 Indeterminate problems.- 12.5 Suppressing low-order modes ..- 12.6 Beams on elastic foundations.- 12.6.1 Axisymmetric buckling of cylindrical shells.- 12.6.2 Whirling of shafts.- 12.7 Energy methods.- 12.7.1 Energy methods in beam problems.- 12.7.2 The uniform beam in compression.- 12.7.3 Inhomogeneous problems.- 12.8 Quick estimates for the buckling force.- 12.9 Summary.- Problems.- A The Finite Element Method.- A.1 Approximation.- A.1.1 The 'best' approximation.- A.1.2 Choice of weight functions.- A.1.3 Piecewise approximations.- A.2 Axial loading ..- A.2.1 The structural mechanics approach.- A.2.2 Assembly of the global stiffness matrix.- A.2.3 The nodal forces.- A.2.4 The Rayleigh-Ritz approach.- A.2.5 Direct evaluation of the matrix equation.- A.3 Solution of differential equations.- A.4 Finite element solutions for the bending of beams.- A.4.1 Nodal forces and moments.- A.5 Two and three-dimensional problems.- A.6 Computational considerations.- A.6.1 Data storage considerations.- A.7 Use of the finite element method in design.- A.8 Summary.- Problems.- B Properties of Areas.- C Stress Concentration Factors.- D Answers to Even Numbered Problems.- Index.

Closed-Form Collapse Moment Equations of Elbows Under Combined Internal Pressure and In-Plane Bending Moment

Abstract: Elastic-plastic finite element analysis has been carried out to evaluate collapse moments of six elbows with elbow factors varying from 0.24 to 0.6. The loading conditions of combined in-plane closing/opening bending moment and varying degree of internal pressure are considered in the analysis. For each case, collapse moment is obtained by twice elastic slope method from the moment versus end-rotation curve. Based on these results, two closed-form equations are proposed to evaluate the collapse moments of elbows under combined internal pressure and in-plane closing and opening bending moment.

Identification of Transverse Crack Position and Depth in Rotor Systems

Abstract: This paper introduces a method for the identification of the position and the depth of a transverse crack in a rotor system, by using vibration measurements As it is reported in literature and from field experience, a transverse crack modifies the dynamic behaviour of the rotor, generating in a horizontal axis shaft periodical vibrations with 1x, 2x and 3x rev components A model-based diagnostic approach and a least-squares identification method in the frequency domain are used for the crack localisation along the rotor The crack depth is calculated by comparing the static bending moment, due to the rotor weight and to the bearing alignment conditions, to the identified 'equivalent' periodical bending moment, which simulates the crack Finally, the validation of the proposed method is carried out statically and dynamically by means of experimental results obtained on a test-rig

Closed-form solution for reinforced Timoshenko beam on elastic foundation

Abstract: This paper suggests a method for obtaining closed-form solutions for a reinforced Timoshenko beam on an elastic foundation subjected to any pressure loading. A particular solution is obtained for uniform pressure loading at any location of the beam. This solution can be used to calculate settlement, rotation, tension, bending moment, and shear force of the beam. A parametric study is carried out to investigate the effects of geosynthetic shear stiffness and tension modulus and the location of the pressure loading. Results are presented and discussed.

Analysis of Continuous Composite Beams Including Partial Interaction and Bond

Abstract: Negative bending moments arise near the intermediate supports in continuous beams; thus, when steel-concrete composite beams are considered, tensile stresses act on the concrete slab. As a result the structural response generally becomes nonlinear due to two main effects: the slip at the slab-profile interface and cracking in the concrete slab. In the present paper a method of analysis for continuous composite beams based on a specific kinematic model of the cross section is proposed. The main feature of the model is the capability to take into account the slip at the slab-profile interface and the slip at the concrete-reinforcement interface. This approach allows the introduction of a constitutive relationship for bond between reinforcing bars and concrete in the theoretical analysis; thus, the tension stiffening effect in the negative bending moment regions can be computed, and the actual mechanical behavior of reinforcing bars of the slab can be analyzed. The results of numerical analyses are compared ...

Asymmetric Four-Point Crack Specimen

Abstract: Accurate results for the stress intensity factors for the asymmetric four-point bend specimen with an edge crack are presented. A basic solution for an infinitely long specimen loaded by a constant shear force and a linear moment distribution provides the reference on which the finite geometry solution is based.

In vitro evaluation of the strength of the conical implant-to-abutment joint in two commercially available implant systems.

Abstract: Statement of Problem: The cone-screw abutment has been shown to diminish micromovement, reducing the burden of component loosening and fracture. However, it is unclear whether the conical taper and joint design influence strength of the interface, with respect to unfavorable bending moments. Purpose: This comparative study evaluated the resistance to bending for the ITI Straumann and Astra Tech ST implant systems using an 8- and 11-degree internal cone, respectively. Material and Methods: Assembled units from each system were mounted in a 3-point bending apparatus. High load tests were performed, 4 mm from the joint, and bending moments necessary to induce first point of plastic deformation and ultimate failure were measured. All units were inspected to determine the critical zone of failure. Results: Bending moments necessary to induce first point of plastic deformation were considered well above that expected in clinical function for both systems. However, the critical zones of failure differed in that the solid Astra abutment deformed before the cone joint with its 11-degree taper and smooth transition into the neck of the screw, preventing screw fracture. By contrast, all ITI screws fractured at the head of the screw where it met the base of the 8-degree cone. It is unclear which aspects of the joint design were responsible for the difference observed in mode of failure or if it was a direct result of the experimental design. Conclusion: For clinically relevant levels of bending moment, no problems were anticipated with respect to component failure for either system. (J Prosthet Dent 2000;83:567-71.)

Spring mounting for an electric generator

Abstract: A support system (50) for an electric generator (10) providing optimal stiffness properties in both the tangential and radial directions The support system utilizes a tapered spring bar (16, 52) having a larger cross-sectional area at points (32) having high bending moment and a smaller cross-sectional area at points (30) having low bending moment By eliminating material in areas of low bending force, a robust support system having increased flexibility is provided The spring bar (17) may further be formed from a plurality of laminations (45) The tapered spring bar (16) may be positioned in a radial direction between the stator core (12) and the stator core frame (13) Alternatively, the tapered spring bar (52) may be connected in a tangential direction between the stator core frame (51) and a spring plate (55) extending longitudinally along a perimeter of the frame (51), with the spring plate (55) being attached to a foundation (57)

Dynamic stability of a rotating shaft driven through a universal joint

Abstract: The dynamic stability of a system composed of driving and driven shafts connected by a universal joint is investigated. Due to the characteristics of the joint, even if the driving shaft experiences constant torque and rotational speed, the driven shaft experiences fluctuating rotational speed, bending moments and torque. These are sources of potential parametric, forced and flutter type instabilities. The focus of this work is on the lateral instabilities of the driven shaft. Two distinct models are developed, namely, a rigid body model (linear and non-linear) and a flexible model (linear). The driven shaft is taken to be pinned at the joint end and to be resting on a compliant damped bearing at the other end. Both models lead to sets of differential equations with time dependent coefficients. For both rigid (linear and non-linear) and flexible models, flutter instabilities were found but occurred outside the practical range of operation (rotational speed and torque) for lightly damped systems. Parametric instability charts were obtained by using the monodromy matrix technique for both rigid and flexible linear models. The transmitted bending moments were found to cause strong parametric instabilities in the system. By comparing the results from the two linear models, it is shown that the inclusion of flexibility leads to new zones of instability, not predicted by previous models. These zones, depending on the physical parameters of the system, can occur for practical conditions of operation. Using direct numerical integration for a few sets of specific parameter values, forced resonances were found when the rotational speed reached a value equal to a natural frequency of the system divided by two.

An Improved Series Expansion of the Solution to the Moving Oscillator Problem

Abstract: The problem of calculating the dynamic response of a one-dimensional distributed parameter system excited by an oscillator traversing the system with an arbitrarily varying speed is investigated. An improved series representation for the solution is derived that takes into account the jump in the shear force at the point of the attachment of the oscillator, which makes it possible to efficiently calculate the distributed shear force and, where applicable, bending moment. The improvement is achieved through the introduction of the quasi-static ' solution, which is an approximation to the desired solution, and is also based on the explicit representation of the solution of the moving oscillator problem as the sum of the solution of the corresponding moving force problem and that of the problem of vibration of the distributed system subject to the elastic coupling force. Numerical results illustrating the efficiency of the method are presented.

Micropolar multiphase model for materials reinforced by linear inclusions

Abstract: A macroscale multiphase model is proposed for assessing the mechanical behaviour of materials reinforced by linear inclusions, such as those commonly employed in geotechnical engineering. The model is developed with the help of the virtual work method and related principles, resulting in the derivation of equilibrium equations and boundary conditions for the matrix and reinforcement phases respectively. The basic concept is the idealization of the inclusions as 1-D-beams continuously distributed throughout the matrix, leading to a micropolar description which accounts for shear force and bending moment densities. The theory includes the possibility of different kinematics for the phases, with non-perfect bonding at the matrix–inclusion interface. Since all the parameters appearing in such a model have a clear mechanical significance, it becomes possible to deal with any boundary value problem involving inclusion-reinforced materials, in a very straightforward manner. Two examples of such problems are solved under the assumption of a linear elastic constitutive law for matrix and reinforcement phases, including their interaction.