Nonlinear Elastic Relation Research Articles

Influence of Ti particle sizes on microstructure and mechanical behaviors of AZ91-Ti composites under different loading paths

Magnesium (Mg) matrix composites are recognized as the most promising light structural materials and have been extensively investigated, but they still suffer from a relatively low ductile nature which seriously affects their further development. Herein, the Mg–9Al–1Zn (AZ91) Mg matrix composites with different sizes of Ti particles reinforcement can be obtained via hot pressed sintering followed by hot extrusion. The microstructure demonstrated that the interfacial region is consisted of Mg17Al12 and Al-rich phase. Uniaxial tensile experiments revealed that the AZ91-Ti (particle size of 15 μm) composite achieves the optimal comprehensive mechanical properties, and the corresponding yield strength (YS), ultimate tensile strength (UTS), elongation (EL) of the composite can reach the high values of 210 MPa, 310 MPa, 15.5%, respectively. Moreover, with the decrease of Ti particle size, grain refinement and the strong interfacial bonding between Ti particles and Mg matrix render as-prepared the composites exhibited outstanding mechanical properties. Ti reinforcement particles of appropriate size can alleviate the plastic strain accumulation during the tensile and compressive cyclic loading, so that the excellent ductility of the composite is obtained. Constitutive relation of AZ91-Ti composites in this experiment can be derived from the nonlinear elastic relation. The simulated stress-strain curves of the composites reinforced by Ti particles with different sizes are consistent with the experiment.

A Molecular Dynamic Study on Nonlinear Vibration Behaviors of Fe Nanowires

In this paper, vibration behaviors of Fe nanowires are investigated by using the large-scale molecular dynamics (MD) simulations. It is observed that the vibration frequency of nanowires rises slightly and nonlinearly with the increase of initial actuation amplitude. Based on the atomic arrangement, a discrete spring-mass model is developed. Its nonlinear elastic relation is used to explain this phenomenon. In addition, Fe nanowires with different lengths and heights show different vibration properties in this work. The ratio between the length ([Formula: see text]) and the height ([Formula: see text]) of nanowires has a significant influence on vibration behaviors. The vibration properties of nanowires can be explained by the Euler–Bernoulli model when the ratio is relatively large, while they can be illustrated by the Timoshenko model when the ratio is relatively small.

Temperature and strain-rate dependent fracture strength of graphene

We investigate the variation in fracture strength of graphene with temperature, strain rate, and crack length using molecular dynamics (MD) simulations, kinetic analysis of fracture with a nonlinear elastic relation, and the quantized fracture mechanics theory. Young’s modulus does not vary significantly with temperature until about 1200 K, beyond which the material becomes softer. Temperature plays a more important role in determining the fracture strength of graphene. Our studies suggest that graphene can be a strong material even, when subjected to variations in temperature, strain rate, and cracks.

Computational Modeling of Damage Growth in Composite Laminates

A progressive damage growth model is developed for composite laminates under compression. The mechanics of damage initiation and growth in a single lamina is modeled in a two-dimensional plane stress setting, using a system of orthotropic nonlinear elastic relations and a set of internal state variables. The latter are associated with different damage mechanisms that are unique to a e ber-reinforced lamina. A thermodynamically consistent set of equations is developed for the evolution of damage growth. The formulation is numerically implemented using the commercially available e nite element package ABAQUS. The present method is applied to analyze the problem of damage growth in a compressively loaded notched laminate. Predictions of the model compared against available experimental observations are promising. EVELOPMENT of computational methodologies for the prediction of progressive damage growth in continuous e ber composite laminates is presently an active area of research. Available predictive methods are based on dee ning strength-based criteria at the lamina level. Based on critical values for tensile, compressive and shear “ strengths,” these methods compute predee ned damage indices that are expressed in functional form in terms of the current stress state. When any of these indices exceeds a predee ned critical value, the material is said to have failed. 1 Beyond initial failure, a consistent and rigorous methodology to account for progressive material deterioration has not been investigated thoroughly. An exception to this is the work by Schapery and Sicking, 2 who carried out lamina-level tests and validated the test results by developing a thermodynamically based progressive damage formation and growth model. In these studies Schapery and Sicking assumed that the e ber-direction response is essentially linear (slight elastic nonlinearity in the e ber direction was accounted for), but damage (microcracking and transverse cracking) formation affected the response in the transverse direction. Consequently, internal state variables that are related to the damage mechanisms in the transverse direction were identie ed, and evolution laws that specify the growth of damage and hence its ine uence on the transverse direction response were prescribed. In contrast to the transverse direction, in which damage accumulation results in progressively decreasing but smooth variations in instantaneous tangent moduli, damage accumulation in the e ber direction leads to nonsmooth abrupt changes in the corresponding moduli. These changes must be properly captured if progressive damage growth in composite laminates is to be modeled accurately. In the present paper Schapery’ s theory (ST) is extended to account for e ber-direction damage (both, in tension and compression) by identifying an additional internal state variable associated with the e ber-direction response. By doing this, we have shown that it is

Nonlinear elastic equation of state of solids subjected to uniaxial homogeneous loading

Applying the finite deformation theory to a solid, which possesses either cubic or isotropic symmetry at stress-free natural state and is subsequently loaded homogeneously in uniaxial direction, one obtains a stress (or strain) dependence of the Young's modulus, Poisson's ratio, and a volume (or density) change, together with a nonlinear elastic relation between stress and strain. These are all expressed in terms of the second and third order elastic constants of the solid material. These expressions are illustrated with examples of cubic silicon crystal, isotropic carbon steel, Pyrex glass, and polystyrene at the relaxed state.

Axisymmetric deformation of shells of revolution made of fiber-reinforced elastomeric layers

The study of the deformation of shells of fiber-reinforced elastomers is based on a structural approach in which the elastomeric matrix and reinforcement are considered separately, without reduction of the composite to a medium having continuously changing characteristics. The concept underlying this approach was first explained in [1], where a momentless model of deformation was used as the basis for describing a method of designing a rubber shell with a reticular reinforcement and no binder. In [2], calculations were performed for certain shells of revolution made of an elastomer and reinforced along the middle surface using fibers with little or no tensility. The effect of the reinforcement was accounted for by a method employed to analyze momentless shells. Line~-r relations from the theory of elasticity were used in [3] to construct a momentless model of the deformation of a shell with a reticular reinforcement with allowance for the effect of the binder. In [4], a method was presented for solving momentless problems of the deformation of multilayered shells of revolution. Here, the mechanical properties of the reinforcing fibers and binders were described with the use of nonlinear elastic relations. The well-known models of deformation do not consider the moment character of the stress state of composite shells made of elastomeric layers. They are suitable for the design of very thin axisymmetric shells, in which the stress state is of a membrane (momentless) character. Here, we construct a structural model that describes the axisymmetric deformation of moment shells of revolution composed of elastomeric layers unidirectionally reinforced with fibers. The method is based on an iterative procedure for the design of shells of this type and employs the hypothesis of a compressible normal element. In accordance with this hypothesis, a normal element of the shell remains straight and normal to the plane of reference during deformation, undergoing elongations that vary through the thickness of each layer. The volume occupied by the reinforcing fibers is assumed to be small compared to the volume occupied by the binder. In accordance with the smallness of the relative volume of the reinforcement, the cross section of the fibers is assumed to be infinitely small. We will use an iterative method to determine the parameters of the deformed shell at each point of discretization of the given boundary-value problem. 1. We will examine a thin composite shell of N elastomeric layers unldirectionally reinforced with fibers. We will assume that the fibers are crossed (arranged antisymmetrically) in any two adjacent layers to form a reinforcing grid with rhomboid cells. The fibers may be arranged in the meridional direction in one layer and in the circumferential direction in the next layer. Some of the layers have no reinforcing material and are purely elastomeric. The thicknesses of the layers is assumed to be constant or to change smoothly in the meridional direction. We take the internal bounding surface of the shell as the plane of reference (the base coordinate plane). We refer this plane to an orthogonal coordinate system established by the lines of principal curvature. The transverse coordinate z is reckoned along an outer normal to the initial reference plane. As the running meridional coordinate, we take the length t of the generatrix of the coordinate surface from a point on the bounding contour to the current point. We will use the following notation: h is the total thickness of the shell; h k is the thickness of the k-th layer; z k and Zk+ 1 are values of the coordinate z for the lower and upper surfaces bounding the k-th layer; k 1 and k 2 are the principal curvatures of the coordinate surface of the shell; r is the distance from a point on the coordinate surface to the x axis (symmetry axis) of the shell; 0 is the angle between the normal to the coordinate surface and the symmetry axis; vkf is the frequency of fibers in the k-th layer; c~kf is the angle of inclination of the k-th layer fibers to the meridian. Quantities pertaining to the reinforcing