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Figure 3. If we utilize Einstein's summation convention, we can leave out the summation symbol and get:. There is a similar process for transforming a second rank tensor, but calculating a formula for the transformation by the same means that we transformed the vector above would be quite laborious. There is a more convenient shortcut. Just as the dielectric constants "maps" the electric field Ej into the electric displacement Di, we can imagine a second rank tensor T kl that takes Q l and produces P k in a given coordinate system:.
We want to find the values for this second rank tensor in a new coordinate system. A valuable tool in tensor math is the identity tensor, which is referred to as the Kronecker delta:. Stress is defined as force per unit area. If we take a cube of material and subject it to an arbitrary load we can measure the stress on it in various directions figure 4. These measurements will form a second rank tensor; the stress tensor. Figure 4. Therefore, it is important to be aware of which sign convention is being used.
A cube with its edges parallel to the principle stress directions experiences no sheer stresses across its faces. Intuitively, this can be seen if one images shrinking the cube in Figure 4 to a point.
If the cube is infinitesimally small, the forces across each face will be uniform. If the cube is to remain stationary the normal forces on opposite faces must be equal in magnitude and opposite in direction and the shear tractions which would tend to rotate it must balance each other. Therefore, it is only necessary to find 6 of the components of the tensor. Important concepts are often used are deviatoric stress and hydrostatic pressure.
Any stress tensor may be broken into two parts. Strain is defined as the relative change in the position of points within a body that has undergone deformation. The classic example in two dimensions is of the square which has been deformed to a parallelepiped. In order for this analysis to work we must only consider infinitesimally small strains. We will call the original length of the side of the square X 1. Therefore we can write:. The Eigen vectors lie in the three directions that begin and end the deformation in a mutually orthogonal arrangement.
If the Eigen vectors are initially of length 1 then in the end they are length:. Strain lends itself well to geometric representation. Think of a unit sphere which has been deformed. Download Free PDF.
A study of lattice elasticity from low entropy metals to medium and high entropy alloys. A short summary of this paper. Abstract An equal-molar CoCrFeMnNi, face-centered-cubic high-entropy alloy system and a face-centered-cubic stainless steel described as a medium-entropy system, are measured by in-situ neutron-diffraction experiments subjected to continuous tension at room and several elevated temperatures, respectively.
With spallation neutron, the evolution of multiple diffraction peaks are collected simultaneously for lattice-elasticity study. Temperature variation of elastic stiffness of a single face-centered-cubic-phase Ni and a single face-centered-cubic-phase Fe are compared as low-entropy metals. Moreover, several nontrivial mechanical behavior of this kind of HEA is observed by Yeh et al. Besides its practical application [1], the underneath mechanical mechanisms of this CoCrFeMnNi-high-entropy alloy are unclear.
For example, how equiatomic-element mismatch affect its bonding [5]? For metallic systems, numerous work have demonstrated the importance of ordering for compositions [7, 8]. Generally, the energy-driven moduli of the soft materials are smaller than their counterparts. The moduli of soft materials typically show temperature-dependent entropy-driven and energy-driven elastic behavior.
For the configuration ordering, the entropic elasticity originates from the thermodynamics. When there is configuration rearrangement, the enthalpy of phase transition can well quantify the level of entropic effect after Clausius—Clapeyron relation [].
For the conformation ordering, spatial distribution of a cluster of atoms or molecules brings significant entropy-driven elasticity to local potential energy, too.
Falk and Langer postulate the dynamics of two-state shear transformation which irreversible motions are governed by local entropic fluctuations in the volumes of the transformation zones at temperatures far below the glass transition. Above all, several results have reported the entropic effect on the elasticity of the metallic systems in terms of the long-range-order-disorder transition [19, 20], martensitic transformation [22], viscoplasticity [21], shear transformation [21], and other plasticity-related behavior [23].
Finally, Hufnagel et al. Wagner et al. Their results show that the local potential energy can vary significantly in space for non-crystalline materials [24, 25]. For long-range order existing in space, a unified elastic modulus is expected [25]. Based on such a fact, for polycrystalline metals, the Taylor theory assumes that the strain tensor is symmetrical [26]. Similarly, Molinari et al. In this paper, we focus on hkl-dependent elasticity that is the consequence originated from the aforementioned effects.
Otto et al. Specifically, Gali and George discovered that the transition between the thermal and athermal regions of the HEAs is higher than that of typical binary alloys [28].
Yeh et al. The extended X-ray absorption fine structure evidence shows that the lattices of HEAs are distorted by the heterogeneity of local chemical distributions [30]. The aforementioned recent research lead us to ask: could these local compositional heterogeneity of the equiatomic high-entropy alloy be sufficiently enough to break tensor symmetry and collaborate with energy-driven elasticity?
We facilitate the advantage of the in-situ neutron diffraction measurements for its great gauge volume to investigate the ensemble-averaged lattice elasticity of the HEA and the MEA to compare with that of the LEMs. The procedures of in-situ diffraction experiments is described in the session of Methods, which is similar to Wu et al. The difference is that besides room temperature experiments, we measure both the HEA and MEA at several elevated temperatures as shown in the Supplementary Information.
The feature is especially useful for the current investigation to observe dynamics loading condition [32]. Moreover, the advantage of the spallation-neutron-source materials-engineering diffractometer, such as VULCAN [32] of SNS, is that multiple diffraction peaks are collected simultaneously. We follow Wu et al. The definition of Ahkl is shown below. Figures 1 c and 1 d are our experimental results. Hutchings et al. Within elastic-deformation limit, there is no plasticity-induced heterogeneous-intergranular stresses.
Besides Hutchings et al. There is no difference between pure Ni and Pure Fe from and lattice moduli as well as their and It is also not obvious for L stainless steel as shown in Figure 1 c.
Meanwhile, for our CoCrFeMnNi HEA, there are no superlattice peaks showing complete-ordered structure throughout the measurements at various temperature. To further examine the aforementioned phenomenon, we revisit our data and plot the results in Figure 2. Good reported such a moduli transition in the Beta-Brass single crystals which is subjected to an order-disorder transition [39]. However, here is neither phase-transformation-related nor order-disorder-transition-related diffraction-peak evolution seen during our measurements of the current CoCrFeMnNi high-entropy alloy.
Rollett et al. The results of HEA is very different to the others as exampled by Lugovy et al. Denton and Ashcroft have discussed the mixture effect on the alloys [6]. They summarize four physical factors including 1 relative atomic sizes; 2 valence electron density; 3 Brillouin-zone effect, and 4 electrochemical differences.
The radius ratio between Cr, Mn, Fe, Co, and Ni in terms of atomic, ionic, covalent, and crystal radii indeed are heterogeneous. Although the atomic numbers of the aforementioned five elements are next to each other, the differences of sizes and individual pair of ionic character do vary, which might correlate with the complicated effect from d-valence electrons.
They propose the bonds have natural lengths L0ij with spring constants Kij. They consider a distribution of random or correlated, of two kinds of bonds. Their model can apply to the current studied face-centered-cubic structure. Similarly, L0B and KB are for correlated bonds, respectively, with probability x.
Referring to the design of the metallic alloys [], the equal-molar CoCrFeMnNi, single face-centered-cubic phase high-entropy alloy can bring in local entropy-driven to the ensemble energy-driven elasticity for the metals. Although experimental results for entropy-driven elasticity is not trivial, hkl-dependent lattice-elasticity has been simulated [50].
Above all, within elastic deformation range, the metals in a long-range-ordered structure are rarely seen to exhibit such an asymmetric tensor responses as this CoCrFeMnNi high-entropy alloy does at K and K.
Within K to K, a crossover of entropy-driven and energy-driven elasticity is observed for the high-entropy alloy. At higher temperature, the severe-distorted-lattice-induced crossover is released. At room temperature, the phenomena is not obvious.
In summary, lattice elasticity of a CoCrFeMnNi high-entropy, a L stainless steel medium-entropy alloys, and a pure Ni and a pure Fe low-entropy metals is investigated. Orientation-dependent hkl-moduli are calculated. The long-range-order face-centered-cubic structure of the high-entropy equal-molar CoCrFeMnNi alloy has never changed throughout the measurements.
Since there is also no appearance of superlattice peaks during the deformation subjected to temperature changes, there is no order-disorder transition within the high-entropy alloy. The most possible mechanism is the competition between the entropy-driven and the energy-driven elasticity originating from the local heterogeneity.
References [1] B. Gludovatz, A. Hohenwarter, D. Catoor, E. Chang, E. George, R.
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