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The growth and coalescence of microvoids nucleating in the second phase particles are the dominant mechanism of ductile fracture. The ductile fracture process is strongly influenced by the stress state. Based on a triaxiality dependent cohesive zone model, the ductile fracture process of zirconium alloys under different stress states is described in the present study. Under the condition of plane strain, a compact tension analysis configuration is established for hydrided zirconium alloys composed of matrix and hydrides. By comparing our prediction with the results based on the extended finite element method, we can calibrate model parameters and then verify the model. The results show that the presence of hydrides accelerates the crack propagation and decreases the post-peak load level. Moreover, we find that the fracture resistance of zirconium alloys is strongly affected by the length, arrangement, quantity, and spacing of the hydrides. Specifically, for hydrides with the length along the crack propagation path, the increase in their length enhances the peak load and reduces the corresponding boundary displacement. The increase in their quantity reduces the post-peak load level. Besides, the increase in their spacing enhances the boundary displacement corresponding to the sudden load drop. For the ductile fracture of zirconium alloys, these simulation results provide insights into their ability to resist crack propagation.

Several recent experiments have revealed the remarkable influence of temperature and graphene concentration on the effective electrical properties of graphene–polymer nanocomposites, but no theory seems to exist at present to quantify such dependence. In this work, we develop a novel micromechanics-based homogenization scheme to connect the microstructural features of constituent phases to the temperature-dependent macroscopic conductivity and permittivity for the nanocomposites. The key microstructural features include the graphene volume concentration, temperature-dependent electrical properties of constituent phases, percolation threshold, imperfect mechanical bonding effect with temperature-degraded interlayer, and the temperature-dependent electron tunneling and Maxwell–Wagner–Sillars polarization. We consider the activation of free electrons and polarization of molecules to write the constitutive equations of polymer, and the collision and vibration probabilities to write those of graphene. We highlight the developed theory with a direct comparison to the experimental data of rGO/epoxy nanocomposites over the temperature range from 293 to 353 K. It shows that before the percolation threshold, the effective electrical conductivity and dielectric permittivity markedly increase with temperature, but after the percolation threshold, the influence of temperature diminishes significantly. In the latter case, the effective permittivity increases only slightly, while the conductivity exhibits an opposite trend.

Bimodal nanostructured (NS) metals possess both ultrahigh strength and good ductility. It is the nanograined (NG) matrix phase that leads to their ultrahigh strength and the coarse-grained (CG) inclusion phase that renders their good ductility. But the overall strength and ductility can also be significantly affected by the behavior of the interface regions. In this study, we employ a cohesive finite-element method to investigate the tensile fracture process of the bimodal NS Cu that includes the interface effects. We develop three types of cohesive elements in the bimodal NS Cu: (i) cohesive elements in the CG phase, (ii) those in the NG phase, and (iii) those at the CG–NG interface. Our objective is to uncover how the strength and ductility of the bimodal NS Cu can be affected by the interface property. In this process, we will also examine how the distribution and shape of the CG inclusions can contribute to the variation of the tensile fracture behavior of the bimodal NS Cu. By an extensive simulation, we find that, even at the small ratio of 1.6% of interface cohesive elements to all cohesive elements, a small change in the cohesive strength of interface elements could lead to a significant change in the overall strength and ductility. We also find that, when the cohesive strength of interface elements exceeds a certain level, the strength and ductility of the bimodal NS Cu will reach a saturation state.

A new category of structural metals with high strength and good ductility is coarse-grained metals strengthened by nanotwinned (NT) regions. This unique combination makes them particularly suitable for applications in bulletproof targets. The extent of this improvement, however, depends on the volume fraction of the NT regions and multiple other microstructural features. Here, a numerical model based on the strain gradient plasticity and the Johnson–Cook failure criterion is developed to study the effects of these attributes. The ballistic performance is quantified by two indices: the limit velocity that measures its ability to resist failure and the limit displacement that measures its ability to resist deformation. The results obtained indicate that, in general, a relatively small twin spacing and a moderate volume fraction of NT regions can achieve both excellent limit velocity and limit displacement. Moreover, array-arranged NT regions are more effective than staggered NT regions in enhancing the limit velocity, but the influence of the array group tends to depend more on the volume fraction of NT regions than the latter one. Mechanism analyses based on the three stages of the impact process and two categories of low-speed impact are performed. The simulated results could provide significant insights into the NT structures for a superior class of nanotwin-strengthened metals for ballistic protection.

The frequency dependence of the electromechanical response of a barium titanate nano-thin film was studied through phase-field simulation. A two-dimensional phase-field model based on Landau–Devonshire energy density function was established in this work. The time-dependent Ginzburg–Landau equation was utilized to calculate the dynamics of the microstructure upon the application of an AC electric field. A segment of barium titanate thin film was modeled with 20 nm in thickness and 80 nm in width. Periodic boundary conditions were applied to both ends of the nano-thin film to represent an infinite length-to-thickness ratio. It was observed from the phase-field results that the loading frequency of the electric field can noticeably affect the hysteresis and butterfly loops of the nano-thin film through competition with the electric dipole evolution. A high-frequency electric field tends to yield a close-to-linear response of the thin film. Meanwhile, it was discovered that the existence of $$180^{\circ }$$ domain walls and their dynamics (oscillation) within the thin film have remarkable influence on the overall response.

Ferroelectric crystals represent a very unique class of multifunctional materials. In addition to strong electromechanical coupling, there exist ferroelectric domains, which can be switched through the application of an electric field or mechanical stress. In addition, these crystals possess several distinct crystal structures over a wide temperature range. As such, phase transition can take place as the crystals are cooled down or heated up, without or with the additional effect of stress or electric field. Domain switch and phase transition represent the two fundamental processes that can affect their microstructures and electromechanical characteristics. In this lecture, we highlight the applications of micromechanics to bulk ferroelectrics and phase fields to nano-structures. The starting points of micromechanics are crystal structures and the Eshelby mechanics, whereas those of the phase fields are the time-dependent Ginzburg–Landau kinetic equation and the Landau–Ginzburg–Devonshire energy density function. We explain how micromechanics can have wide applicability in the study of domain switch and phase transition, and change of dielectric constants, of bulk BaTiO3 crystals, and how phase fields can provide the nano-scale domain patterns, influence of surface tension on free-standing BaTiO3 nano-thin films, and grain-size dependence of ferroelectric characteristics in nano-grained BaTiO3 polycrystals.

In a nanocrystalline solid a significant portion of atoms resides in the grain boundary and the nearby outer grain. This combined region, known as the grain-boundary affected zone (GBAZ), is plastically softer than the grain interior, and it are the combined contributions of the grain interior and GBAZ that give rise to the overall response. In this spirit a two-phase composite model is developed to study the high-temperature creep resistance of nanocrystalline materials. Here the rate equation of each phase is represented by a power law and the Arrhenius function, but that of the grain interior is further taken to scale with the Hall–Petch relation whereas that of the GBAZ remains independent of grain size. This unified constitutive equation in turn leads to the concept of secant viscosity. Then a homogenization theory is developed by means of a transition from linear viscoelasticity to nonlinear viscoplasticity with the Maxwell viscosity constantly replaced by the secant viscosity. Subsequently a field-fluctuation method is called upon to determine the effective stress of both phases. The developed theory is applied to model the creep behavior of nanocrystalline Cu, NiP alloy, and Ni at various levels of stress, temperature, and grain size, with results that reflect good agreement with available experiments. We then applied the theory to examine the nature of creep resistance as the grain size decreases in the nanometer range in some detail, and it was discovered that creep resistance in the Hall–Petch like plot undergoes a transition from a positive slope to leveled off, and then to a negative slope. The leveled-off value in effect represents the maximum creep resistance that a material can attain, and it is found that this occurs at a critical grain size, d crit, that exists in the nanometer range.

This paper considers the effective bulk moduli and the underlying elastic fields of a particle- and a fiber-reinforced composite whose matrix properties are graded linearly along the radial distance. The governing Fuchsian equations are first transformed into Riemann equations which, by means of change of the dependent variable, are further transformed into hypergeometric equations which are solved in terms of the hypergeometric function. This set of solutions provides an explicit dependence of the effective bulk behavior of the composites with arbitrary variations of the slope of the matrix moduli.

In this paper we take a fresh look at the classical problem of Hill's arithmetic and geometric means. It is shown that both types of means can be derived from a dual class of polarization fields that lead to an “ordered” structure. Hill's arithmetic mean of Voigt's and Reuss' moduli is derived from the displacement-prescribed boundary condition, whereas the arithmetic mean of the compliances is derived from the traction-prescribed condition. These means are found to correspond to 1/4-th of the maximum elastic energy from the perturbed elastic field, the maximum values being associated with the Reuss and Voigt approaches, respectively. Hill's geometric mean, on the other hand, always lies between this pair of moduli. It also represents the limiting case of the arithmetic means for this class of ordered materials that can be constructed with a hierarchical scheme. Despite their simplicity, both the arithmetic and the geometric means are found to fully meet the rigorous requirement for the shift property of the effective moduli recently derived by Hu and Weng [10].

Based on the principle of thermodynamic equilibrium, the condition of stress-induced phase transformation in a two-phase ceramic is established. The development makes use of the change of potential energy that was calculated with a mean-field approach. In this process the elastic heterogeneity of the constituent phases, and the shape and volume concentration of the randomly oriented metastable ellipsoidal inclusions, are fully accounted for. Both the transformation heightH of the process zone with a steadily growing crack and the fracture toughness increment ΔK of the transforming system are derived. The derived theory is then used to address the effect of inclusion shape and elastic inhomogeneity on the transformation toughening of two-phase ceramics. By considering the metastable ellipsoidal inclusions as phase 1 and the stable matrix as phase 0, it is found that, when μ1/μ0>1, flat-like discs always provide a larger transformation-height while spherical ones provide the smallest, and vice versa. As the ratio of μ1/μ0 increases, the size of the process zone also increases. For the toughness increment, the results indicate that thin-disc inclusions are again the most effective toughening medium. It is further found that Poisson's ratio of the constituent phases also has a significant effect; the combination ofv1→0.5 for the inclusions andv1→0 for the matrix has the best enhancement for fracture toughness. But whenv1→∞, the toughness increment ΔK all approaches to an asymptotic value regardless of the values of Poisson's ratios. Some explicit solutions of toughness change for several distinctive shapes of inclusions are also derived for the first time.

The dynamic response of a finite crack in an unbounded Functionally Graded Material (FGM) subjected to an antiplane shear loading is studied in this paper. The variation of the shear modulus of the functionally graded material is modeled by a quadratic increase along the direction perpendicular to the crack surface. The dynamic stress intensity factor is extracted from the asymptotic expansion of the stresses around the crack tip in the Laplace transform plane and obtained in the time domain by a numerical Laplace inversion technique. The influence of graded material property on the dynamic intensity factor is investigated. It is observed that the magnitude of dynamic stress intensity factor for a finite crack in such a functionally graded material is less than in the homogeneous material with a property identical to that of the FGM crack plane.

For a two-phase isotropic composite consisting of an isotropic matrix and oriented isotropic ellipsoidal inclusions, Mori-Tanaka's (MT) [6] method and the more recent Ponte Castaneda-Willis (PCW) [1] method are perhaps the only two methods that deliver explicit results for its effective moduli. An attractive feature of the MT method is that it always stays within the Hashin-Shtrikman [3] bounds, while the novel part of the PCW approach is that it has a well defined microstructure. In this paper, we made a comparative study on these two models, for both elasticity and their applications to plasticity. Over the entire range of inclusion shapes, the PCW estimates are found to be consistently stiffer than the MT estimates. An investigation of the possibility of a PCW microstructure for the MT model indicates that the MT moduli could be found from the PCW formulation, but this would require a spatial distribution that is identical to the oriented inclusion shape. Such a requirement implies that the underlying two-point joint probability density function is not symmetric, and thus it is not permissible. One is led to conclude that, unlike the aligned case, the MT model cannot be realized from the PCW microstructure with randomly oriented inclusions.

Summary. A field-fluctuation method is introduced into the secant-viscosity framework to evaluate the homogenized effective stress of the heterogeneously deformed elastic-viscoptastic matrix in an isotropic composite. Two microgeometries are considered here: one is reinforced with spherical particles and the other with randomly oriented thin discs. The time-dependent creep strains of the elastic-viscoplastic composites are then calculated as a function of inclusion volume concentration. As these two microgeometries are known to provide the lower and upper bounds of the effective moduli in elasticity, the creep strains associated with these two inclusion shapes are believed to set the upper and lower ranges of the overall creep strain for all inclusion shapes. Detailed comparison with a previously developed direct work-rate method is also made. While the effective stress of the ductile matrix - and therefore the overall creep strain of the composite - are higher by the direct work-rate method, the difference between the two is found to be small. However, when the Laplace inversion of the effective secant viscosities of the viscoelastic comparison composite can be cast in an explicit form, the field-fluctuation method will provide a complete evaluation of the effective stress, and is also substantially simpler than the direct work-rate method.

A simple constitutive equation involving only six material constants has been developed to describe the high temperature deformation of metals over a wide range of stress and temperature. The model involves a drag stress and a back stress, along with their evolution equations. A new concept has been introduced to connect the growth behavior of these two internal state variables so as to reduce the number of material constants required in the description under cyclic loading conditions. It is demonstrated that this simple model can adequately describe the stress-strain behavior of a 617 Nickel-base superalloy over three different orders of strain rate, and up to 900°C. It can also provide the cyclic hardening behavior and stable hysteresis loop under a strain-controlled loading, and the strain-ratchetting under an off-axis, stress-controlled loading, among others. The influence of stress and temperature on creep, and the influence of mechanical and thermal cycling on the creep strain accumulation can all be studied with this set of simple constitutive equations.

The overall elastic moduli and elastoplastic stress-strain relations of a fiber-reinforced composite containing either aligned or randomly oriented spheroidal voids are derived explicitly. The results are given in terms of the void shape and concentration, and for the elastic moduli the void shape is further pushed to the limit for cracked bodies. The present theory also corresponds to the condition of “random bridging”, with a longitudinal Young's modulus and axial shear modulus lying between those of “full bridging” and “no bridging”. Numerical results further indicate that both elastic and elastoplastic strength of the fiber-reinforced composite can be significantly weakened by the presence of both types of distribution, but the extent of stiffness or strength reduction is highly dependent upon the void shape and loading direction.

The influence of ellipsoidal inclusions and elliptic cracks on the overall effective moduli of a two-phase composite and of a cracked body, respectively, is investigated by means of Mori-Tanaka's theory for three types of inclusion and four types of crack arrangements: monotonically aligned, 2-D randomly oriented (two kinds for cracks), and 3-D randomly oriented. The effective moduli of the composite in the aligned case are known to coincide with Willis' orthotropic lower (or upper) bounds with a two-point ellipsoidal correlation function if the matrix is the softer (or harder) phase. With 2-D randomly oriented inclusions, the effective moduli are examined under Willis' transversely isotropic bounds with a two-point spheroidal correlation function, and it is found that, as the cross-sectional aspect ratio of the ellipsoidal inclusions flattens from circular shape to disc-shape, the two effective shear moduli and the plane-strain bulk modulus all lie on or within the bounds. The effective bulk and shear moduli of an isotropic composite containing randomly oriented ellipsoidal inclusions also fall on or within Hashin-Shtrikman's bounds as the shape of the ellipsoids changes. The obtained moduli are then extended to a cracked body containing elliptic cracks, which are generated by compressing the thickness of ellipsoidal voids to zero. It is found that only selected components of the effective moduli are dependent upon the crack density parameter η. Their dependence on η and the crack shape γ are explicitly established.

At any generic state, the tangent moduli and compliances of a metal crystal are derived in terms of its elastic moduli and compliances, and its physical slip system hardening modulih ij. The structure ofh ij is explored in conjunction with a mixed hardening law. It is found that the latent hardening moduli (h ij,i≠j) are related to the active hardening moduli (h ij,i+j) through the latent hardening coefficients, and that each active hardening modulus is composed of the selfhardening, single slip modulush and the latent structural-change hardening modulih ij ′ . The theory is supplemented with some suggested functions forh andh ij ′ , suitable for metal forming analysis. The derived constitutive relations are finally applied to calculate the tensile stress-strain relations of aluminum and zinc crystals under finite strains.

The effective elastic moduli for a class of porous materials with various distributions of spheroidal voids are given explicitly. The distributions considered include the unidirectionally aligned voids, three-dimensionally and two-dimensionally, randomly oriented voids, and voids with two types of biased orientations. While the 3-d random orientation results in a macroscopically isotropic solid, the porous media associated with the other arrangements are transversely isotropic. The five independent elastic constants for each arrangement, as well as the two for the isotropic case, are derived by means of Mori-Tanaka's mean field theory in conjunction with Eshelby's solution. Specific results for long, cylindrical pores and for thin cracks with the above orientations are also obtained, the latter being expressed in terms of the crack-density parameter. Before we set out the analysis, it is further proven that, in the case of long, circular inclusions, the five effective moduli of a fiber composite derived from the Mori-Tanaka method coincide with Hill's and Hashin's lower bounds if the matrix is the softer phase, and coincide with their upper bounds if the matrix is the harder.

With the current state as reference, the initial and subsequent yield surfaces of a single crystal are constructed under finite deformation in both Kirchhoff stress and its conjugate strain space. In both cases the yield condition of a slip system is characterized by a quadric surface rather than by a linear yield plane. As consequences of Drucker's and Ilyushin's postulates, these quadric surfaces are necessarily convex; so is the yield surface of the crystal. The translation and rotation of each quadric surface and the distortion of the yield surface are determined following an incremental loading. The loading criterion, its associated flow rule and the normality structure in connection with a yield surface at finite strain are also rigorously analyzed from physical grounds.

A micromechanical theory is developed to study the time-dependent creep behavior of a two-phase alloy, consisting of a polycrystalline matrix and uniformly dispersed spherical particles. It is shown that there are two major sources of dispersion hardening-metallurgical, involving the bypassing of particles by dislocations, and mechanical, concerning the stress redistribution among the constituent phases and the calculation of creep strains. The metallurgical effect is implemented in a dispersion-dependent micro constitutive equation of slip systems. The problem of stress redistribution due to both the elastic and the plastic inhomogeneity is also analyzed. The results indicate that, while the stress in the creeping matrix of a constituent grain tends to decrease, the stress in the embedded inclusions is dependent upon the grain orientation in which they are embedded, and may increase or decrease. The overall creep behavior is calculated from the averaging process over the orientation of its constituent grains. This theory is finally applied to a cobalt system reinforced with rutile particles. Some merits and limitations of the theory are also discussed.