![]() If a pair of elastic moduli is known, all other moduli can be derived by calculation. Therefore homogeneous and isotropic solid materials can be characterized by linear elastic properties, which are fully described by two major elastic components that are any pair of moduli. The Lame constants can be calculated from the experimentally determined elastic modulus. Here, E is Young's modulus, K is the bulk modulus, ν is Poisson's ratio, and G is the shear modulus. Young's modulus is measured in GPa-for example, for aluminum, E = 70 GPa for iron, E = 180 GPa, but the largest Young's modulus is seen for graphene, where E = 1000 GPa. In cubic crystals, E modulus equates to three diagonal components of elastic stiffness that are identical: E = c 11 = c 22 = c 33. In simple cases, Young's modulus is defined as the ratio of stress to elongation. The modulus of elasticity is a set of physical quantities that characterize the ability of any solid body to be elastically deformed under conditions where force is applied to it. Young's modulus ( E) or modulus of longitudinal elasticity describes a material's resistance to stretching or compression during elastic deformation. Shear modulus is expressed by the ratio of shear stress to shear strain that is defined as the alteration in the right angle between planes, whereon shear stresses are applied to two mutually orthogonal sites. The shear modulus, or modulus of rigidity (abbreviated as G or μ), characterizes the stressed state in case of net shear, that is, the ability of a material to resist any change in its shape while maintaining its volume. These ratios that reduce to six independent components of elastic stiffness are the Cauchy relations. ![]() In these materials, upon application of a stretching force, the transverse section of the body increases.Ĭ 23 = c 44 c 13 = c 55 c 12 = c 66 c 14 = c 56 c 25 = c 46 c 36 = c 47. There are materials (polymers) for which Poisson's ratio is negative these materials are called auxectics. For example, most steels have v ~ 0.3 for germanium, v = 0.31 for quartz glass, Poisson's ratio is small ( v = 0.17), whereas for rubber, Poisson's ratio is large: v ~ 0.6 ( ν is measured in relative units: mm/mm, cm/cm, etc.). In case of an entirely brittle material, the Poisson ratio is zero, whereas for a completely elastic material, ν = 0.7. Its value is the ratio of the linear contraction of cross-section e' to the elongation e, that is, ν = | e'|/ e. Poisson's ratio shows how the cross-section of a deformable body changes under lengthwise stretching (or compression). During this stretching, in the vast majority of cases, the cross-section of the material decreases. When a stretching force is applied lengthwise to a solid, the solid starts to stretch. Poisson's ratio ν is often used to characterize the elastic properties of a material. It should be noticed that the value of the transverse strain measured might be more accurate and stable if a circumferential extensometer was used instead of a strain gage. This could be attributed to the bridging effect of fibers acting as crack arrest in the matrix, which could restrict the lateral expansion of the specimen. In contrast, Poisson's ratio of ECC kept nearly constant until extremely high section stress was achieved (i.e., 90% of the peak stress), which was consistent with the results obtained by Zhou et al. However, as the normalized section stress surpassed 0.7, Poisson's ratio of concrete increased exponentially, which was caused by the accelerated crack propagation in concrete, which led to an obvious transverse expansion of the specimen. It is found that for ordinary concrete, Poisson's ratio stayed almost stable when the normalized section stress was less than 0.7. The compressive strength of ordinary concrete was 80 MPa. The relationships between Poisson's ratio and the normalized section stress before peak load of PE-ECC and ordinary concrete ( Elaqra et al., 2007) are compared in Fig.
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