Search

Overall mechanical properties and local strain and stress field averages, caused in individual phases of heterogeneous solids by remotely applied uniform strain or stress, are often derived from estimates of local fields in ellipsoidal homogeneous inclusions and inhomogeneities, bonded to a large volume of a surrounding matrix or ‘comparison medium’. The attraction of this approach lies in the relative simplicity of evaluation of the local fields, and in the adaptability of ellipsoidal shapes, such as prolate or oblate ellipsoids, spheroids, cylinders, spheres, penny-shaped discs or slits, to represent either short or long fibers, particles, voids and cracks of different shapes. Transition from local fields in a single inhomogeneity to those in interacting inhomogeneities comprising composite aggregates and polycrystals is accomplished, in part, by assigning certain properties to the comparison medium, as shown in Chaps. 6 and 7.

Together with the methods described in the previous chapter, overall moduli and local field averages in the phases can be estimated by one of several approximate methods, which use different models of the microstructure. Among those described here are variants of the average field approximation, or AFA, which rely on strain or stress field averages in solitary ellipsoidal inhomogeneities, embedded in large volumes of different comparison media L 0. Among the most widely used procedures are the self-consistent and Mori-Tanaka methods, and the differential scheme, described in Sects. 7.1, 7.2 and 7.3. Those are followed by several double inclusion or double inhomogeneity models in Sect. 7.4, and by illustrative comparison with finite element evaluations for functionally graded materials in Sect. 7.5.

This chapter is concerned with composites and polycrystals, consisting of two or more distinct phases that have known stiffnesses L r defined in the fixed overall coordinate system of a representative volume V. Phase volume fractions C r \( \Sigma_{{r = 1}}^n\,{c_r} = 1, \) are no longer small, hence evaluation of both overall properties and local fields must reflect interactions between individual phase volumes. Spatial distribution of the phases in V is statistically homogeneous, as described in Sect. 3.2.2, and perfect bonding is assumed at all interfaces. Of interest are derivations of upper and lower bounds on the overall stiffness \( {L} = {{L}^{\text{T}}} \) and compliance \( {M} = {{L}^{{ - 1}}} \) of the aggregate, and of estimates of phase volume averages of strain and stress fields, caused in the heterogeneous system by application of uniform overall strain \( {{\varepsilon }^0} \) or stress \( {{\sigma }^0} \). Those are sought in terms of known volume fractions, elastic moduli, shape and alignment of the constituent phases, Sects. 6.1 and 6.2.

Derivations and presentations of results in this book will appear in the tensor components, or in the related matrix notation. In the tensor component or subscript notation, vectors or first-order tensors are denoted by lower case italics with a single letter subscript, such as \( {n_i} \) or \( {\nu_j} \), while second, third and fourth-order tensors are written as \( {\varepsilon_{\textit{ij}}},\,\,{ \in_{\textit{ijk}}},\,\,{L_{\textit{ijkl}}} \), with the number of subscripts indicating the order or rank R of the tensor. The subscripts have a certain assigned range of values, which is i, j,… = 1, 2, 3, or \( \rho = 3 \) for tensorial quantities in the Cartesian coordinates \( {x_i} \). The number of tensor components is \( N = {R^{\rho }} \). It is then convenient to write the components of a first, second or fourth order tensors as \( {(3} \times {1),}\,\,{(3} \times {3)}\,\,{\text{\; or (9}} \times {9)} \) arrays, which need not conform to the rules of matrix algebra. The third order tensor can be displayed in three \( {(3} \times {3)} \) arrays.

Several types of bonds may exist at the juncture between adjacent phases in contact. On the microscale of many composite materials, most desired is a perfect bond along a sharp spatial boundary S of vanishing thickness. It guarantees that both traction and displacement vectors remain continuous on S. Contact between phase surfaces may also involve presence of one or more interphases, thin bonded layers of additional homogeneous phases introduced, for example, as coatings on particles or fibers, or as products of an interfacial chemical reaction. During composites manufacture and/or loading, an interface is expected to transmit certain tractions between adjacent constituents. When the resolved tensile and/or shear stress reaches a high magnitude, the interface may become imperfect by allowing partial or complete decohesion, a displacement jump, possibly accompanied by a distribution of ‘adhesive’ tractions. In an opposite situation, a high compressive stress may cause radial cracking in one of the phases in contact, or in the surrounding matrix. While magnitudes of interface tractions determine material propensity to distributed damage, the work required by either decohesion or radial cracking must be provided by release of potential energy, which is proportional to phase volume Chap. 5. Therefore, small inhomogeneities are less likely sources of damage than large ones.

This chapter provides a short introduction to constitutive relations for materials that exhibit incremental elastic-plastic deformation in response to an applied loading path which extends beyond their initial yield surface. In a certain sense, it is analogous to Chap. 2 on Anisotropic Elastic Solids, with which it shares the results pertaining to isotropic elasticity. Moreover, the instantaneous tangential stiffness or compliance matrices may have as many as 21 nonzero coefficients, as in triclinic elastic materials. In preparation Chap. 12, attention is focused on those parts of incremental plasticity theory that are useful in modeling of metal matrix composites.

As in the previous chapter, we consider homogeneous inclusions and inhomogeneities in subvolumes \( {\Omega_r} \) of an infinitely extended homogeneous volume \( {\Omega_{{0}}} \) of a comparison medium or ‘matrix’ of stiffness \( {{L}_0} \); the total volume \( \Omega = {\Omega_{{0}}} + {\Omega_r} \). In Sects. 5.1.4 and in 5.2 and 5.3, we examine composite aggregates with dilute reinforcement, which may consist of many distinct inhomogeneities \( {{L}_r} \) in a matrix \( {{L}6557} \), as described in Sect. 4.4. Systems containing cracks are discussed in Sect. 5.4. Loads applied to both single and multiple inhomogeneity systems include displacement or traction fields acting at a remote boundary to generate uniform overall strain or stress, and piecewise uniform, physically based eigenstrains in both matrix and inhomogeneities. Those include thermal and moisture-induced strains, phase transformations, and inelastic strains. Low loading rates causing only small strains are assumed.

This chapter provides a brief introduction to micromechanics. Following an overview of several descriptors of microstructural geometry is an outline of the procedures that predict overall response of a heterogeneous aggregate in terms of phase volume averages of local strain or stress fields. Applied loads include uniform overall strain or stress and a piecewise uniform distribution of eigenstrains or transformation strains in the phases. Derivations of theorems, formulae and connections that will frequently be used in subsequent chapters are presented in Sects. 3.7, 3.8 and 3.9. A summary of the overall and local response estimates appears in the concluding Sect. 3.10. Many symbols used in this and following chapters are summarized in Table 2.5.

Response of conventional and modified sandwich plate designs is examined under static load, impact by a rigid cylindrical or flat indenter, and during and after an exponential pressure impulse lasting for 0.05 ms, at peak pressure of 100 MPa, simulating a nearby explosion. The conventional sandwich design consists of thin outer (loaded side) and inner facesheets made of carbon/epoxy fibrous laminates, separated by a thick layer of structural foam core. In the three modified designs, one or two thin ductile interlayers are inserted between the outer facesheet and the foam core. Materials selected for the interlayers are a hyperelas-tic rate-independent polyurethane;a compression strain and strain rate dependent, elastic-plastic polyurea;and an elastomeric foam. ABAQUS and LS-Dyna software were used in various response simulations. Performance comparisons between the enhanced and conventional designs show that the modified designs provide much better protection against different damage modes under both load regimes. After impact, local facesheet deflection, core compression, and energy release rate of delamination cracks, which may extend on hidden interfaces between facesheet and core, are all reduced. Under blast or impulse loads, reductions have been observed in the extent of core crushing, facesheet delaminations and vibration amplitudes, and in overall deflections. Similar reductions were found in the kinetic energy and in the stored and dissipated strain energy. Although strain rates as high as 10−4/s1 are produced by the blast pressure, peak strains in the interlayers were too low to raise the flow stress in the polyurea to that in the polyurethane, where a possible rate-dependent response was neglected. Therefore, stiff polyurethane or hard rubber interlayers materials should be used for protection of sandwich plate foam cores against both impact and blast-induced damage.

The Salt tolerance of facultatively halophytic Lophopyrum elongatum and closely related less salt-tolerant wheat Triticum aestivum L. is enhanced if plants are allowed to gradually acclimate to salt stress than if they are suddenly shocked. This acclimation to salt stress is regulated by Abscisic acid (ABA); a pretreatment with ABA substituted for the acclimation and increased tolerance of salt shock. The ABA-induced acclimation is rapid and coincides with enhanced expression of the “Early Salt Induced” genes (ESI) in the roots. Lophopyrum elongatum tolerates salt shock better than wheat and its genome confers greater tolerance of salt shock on wheat in their amphiploid. The tolerance of salt shock is principally controlled by chromosome 3E in the Lophopyrum elongatum genome. Wheat homoeologous chromosome 3A and 3D also control salt shock response. It is speculated that chromosome 3 in both species mediates salt shock response via ABA.

A new model for incremental analysis of distributed damage evolution in heterogeneous solids is developed with the Transformation Field Analysis. Stress changes caused by local debonding under increasing overall loads are described by a selected model of imperfectly bonded inhomogeneity and represented by equivalent eigenstrains that act together with the applied loads and prescribed local transformation strains on an undamaged aggregate. Interaction between the still bonded and partially debonded phases at any damage state is described by transformation influence functions. Damage rates are derived from the local fields, in terms of a prescribed probability distribution of interface strength and local energy released by debonding. An incremental procedure is outlined that predicts the extent of damage and its effect on local and overall response under variable loads. Also, potential applications of fiber prestress in damage control and prevention in laminated structures are illustrated by two examples. One shows how prestress release can expand the tensile damage-free region of symmetric laminates. Another derives the fiber prestress magnitudes needed to eliminate free edge stress concentrations caused in any laminated plate by cooling from the processing temperature.

Fatigue failure of metal matrix composite laminates is often preceded by a substantial loss of stiffness associated with cyclic plastic straining and subsequent low-cycle fatigue crack growth in the matrix. Experimental observations indicate that two damage patterns evolve under cyclic loading beyond the elastic range, one formed by cracks extending along the fibers in off-axis plies, and another consisting of cracks bypassing the fibers at an angle in axially loaded plies. Damage saturation is observed under constant load amplitudes. Guided by these experiments, the damage evolution process analyzed herein is regarded as a shakedown mechanism, and damage saturation as a shakedown state. For a given program of variable cyclic loading, evaluation of crack densities needed for shakedown is formulated as a nonlinear constraint optimization problem, where the total damage in a laminate is evaluated from the minimization of a cost function that corresponds to a measure of total damage. The associated nonlinear constraints are derived from the ply yield criterion, hardening rule, and physically motivated bounds on the damage parameters. Effective elastic stiffness reduction and local stress redistribution predicted by the optimization procedure are compared with experimental measurements on B/Al laminates.

Fracture of notched, unidirectionally reinforced composite plates with well-bonded ductile matrices is typically preceded by the formation of long, discrete plastic shear zones aligned in the fiber direction. Onset of fracture is associated with a critical tension stress in a certain small process zone ahead of the notch tip; the critical stress is often equal to the tensile strength of the unnotched composite plate. Length of the shear zones can be estimated by plastic limit-analysis, and the local tension stress ahead of the notch found by superposition of the stress caused by remotely applied loads with the stress induced by the shear tractions in the plastic zone, which is shown to be dominated by a logarithmic singularity. In as-fabricated boron-aluminum composites, this fracture mechanism was analyzed and confirmed by numerous experiments (G.J. Dvorak, J. Zarzour and Y. Benveniste, Engineering Fracture Mechanics 42, 501–517, 1992). Since the notch tip field is not described by a stress intensity factor, experimental notched strength data cannot be interpreted in terms of a single material property, such as toughness. An alternative scaling procedure is outlined for prediction of notched strength of wide plates on the basis of data obtained from small size specimens.

Fiber prestraining is often employed to reduce fiber waviness while fabricating composite cylindrical structures by filament winding or fiber placement. The effect of fiber prestraining on the final residual stress state in a closed-end laminated cylinder fabricated on a steel mandrel is analyzed here. The elements influencing the residual stress state are identified and the corresponding influence functions relating the ply stresses to the fiber prestrain are obtained. The ply residual stresses are evaluated for a constant prestrain distribution in all the layers. An optimization procedure is implemented to find prestrain distributions that produces minimal residual stresses in layers. The role of mandrel stiffness on final residual stress state is also studied by selecting different mandrel thicknesses.

After fabrication or in service, certain metal and ceramic matrix laminates may exhibit extensive decohesion, such as debonding and sliding at fiber-matrix interfaces. The evolution of the decohesion process, as well as its effect on overall response and on redistribution of local stress and deformation fields, is analyzed here with a modification of the transformation field analysis method (Dvorak 1992). A finite element analysis of a unit cell model of a fibrous ply is used to obtain the decohesion-induced changes in the potential energy, needed in local fracture criteria. Changes in the local stresses and overall strains are found and converted into effective phase eigenstrains that produce identical average fields in a perfectly bonded ply. Finally, transformation strain analysis of a laminated plate is developed for derivation of governing equations for incremental evaluation of the local fields along a given overall loading path. Applications indicate that the decohesion process may generate overall response similar to that of an elastic-plastic laminate.

A brief survey is presented of recently developed techniques for estimating overall elastic stiffness of statistically homogeneous multiphase solids subjected to large stress gradients. The stiffness estimates, based on linearly varying fields, provide local properties for a finite element analysis of functionally graded composite materials. Comparisons with selected experiments are encouraging, however, significant differences in local fields and overall response are found in comparisons with predictions obtained from standard homogenization techniques.

The transformation field analysis (TFA) of inelastic composite materials (Dvorak 1992) is extended here to fibrous composite laminates. Loading is limited to uniform in-plane stresses and out-of-plane normal stress, and to uniform changes in temperature. The solution for local stresses or strains in the plies is found in terms of elastic transformation influence functions and concentration factors which reflect a selected microgeometry representation of a unidirectional composite, and the constraints imposed on the in-plane strains of the perfectly bonded plies. This methodology is applied in simulations of hot isostatic pressing and subsequent loading of a (0/90)s Sigma/Timetal 21S laminate under axial tension/tension stress cycles applied at constant temperature.

The effect of uniform thermal changes on the position of yield surfaces in the overall stress space is described for two-phase fibrous and particulate composite materials with an elastic-plastic matrix. An auxiliary uniform strain field is used to evaluate this effect in an exact manner. The thermal change is shown to cause a rigid-body translation of all local branches of the yield surface in the overall stress space, in the direction of the overall stress associated with the uniform strain field. In a similar way, the contribution of a thermal change to the overall strain is evaluated from the solution of a mechanical loading problem, in terms of an equivalent overall stress increment, and a uniform isotropic strain change. Such connections between mechanical and thermal loads in inelastic analysis of composites have been described in [1–4]; their applications in processing of composite laminates appear in [5].

Determination of macroscopic mechanical properties of heterogeneous media from the properties of their constituents is a problem of long standing in the mechanics of solids. Current activity in the field includes many areas of applications, such as particulate and fibrous composites, polycrystals, concrete, and geological materials. This lecture first presents a survey of constitutive theories of elastic, anisotropic heterogeneous media, which have been useful in these applications. The second part reviews our recent results in the area of elastic-plastic deformation of laminated metal matrix composite plates. Finally, fracture and fatigue in fibrous composites are briefly discussed.

A criterion for propagation of an existing cleavage microcrack through the grain boundary and on a prospective cleavage plane in the neighboring grain is developed for an arbitrary mutual orientation of the two crack planes. This criterion is then generalized to the case of selective microcrack propagation on the {100} cleavage planes in iron and other b.c.c. polycrystals, which are assumed to have no preferential orientation. A relationship between the magnitude of the applied stresses and the probability of successful grain boundary crossings is derived for the case of a uniaxial tension stress. In this particular case, the relative portion of grains which are affected by propagating microcracks increases in direct proportion to the magnitude of the applied stress until about 60% of all grain boundaries have been penetrated. Accordingly, the polycrystal can be selectively weakened at relatively low applied stresses by cleavage microcracks propagating along a continuous, multiply connected, nearly planar path which leads through grains that have the most nearly normal cleavage planes relative to the stress axis.

Conditions for crack-branching are reviewed and relationships between the stress intensity for crack-branching and fracture toughness derived. For very brittle materials microscopic sub-branching should be observed when the crack tip stress intensity reaches ∼1.4 K Ic but in most metals the stress intensity required will be 2.3 K Ic or greater. Larger stress intensities are required to promote full bifurcation. Experimental confirmation is presented, and some consideration given to stress-corrosion crack-branching, microstructural factors and crack-branching in bursting tubes.