*3.2. Stress Concentration Induced by the State of Crystal Orientation*

The calculated elastic constants *Dijkl* and the aforementioned mechanical properties were used in FEA for a tension load of 50 MPa, which is much less than the yield strength of Ni3Al (>350 MPa [30]). Covalent bonding and ionic bonding are well known to lead to brittleness in intermetallic compounds. Herein, the crystal plane (111) was selected to evaluate the chemical bonds between Ni–Ni and Ni–Al atoms, and the electron density in the (111) planes was calculated via DFT (Figure 2). From the overlap of the electron densities of Ni–Ni atoms, it is inferred that covalent bonds exist. Meanwhile, the electron charge density around Ni atoms is 1.0, whereas that close to Al atoms is 0.0, indicating that the electrons of Al atoms move toward the Ni atoms and that Ni and Al atoms are bonded together by ionic bonds. Therefore, we assumed that Ni3Al exhibits some brittleness.

According to the maximum principal stress theory, in a simple tension test, failure occurs if the first principal stress (σ1) reaches the elastic limit stress. Figure 3a1,a2 display the distribution of σ<sup>1</sup> in the submodel and the intermediate layer, where the σ<sup>1</sup> is maximally localized at the edge of the intermediate layer (refer to the red contour). The maximum σ<sup>1</sup> and average σ<sup>1</sup> are 331.9 and 200.1 MPa respectively, and the ratio of the maximum σ<sup>1</sup> to average σ<sup>1</sup> is 1.66. Clearly, different states of crystal orientation induce stress concentration in the intermediate layer, which may lead to failure in this layer. Moreover, Pugh et al. [31] proposed that the ratio between the bulk modulus and the shear modulus (i.e., *B*/*E*) can be used to evaluate the ductility of a material; they also proposed that ductile materials have *B*/*E* values greater than 1.75. The calculated *B*/*G* ratio for Ni3Al in this study is 2.37; we therefore speculate that Ni3Al also has some ductility, which can also be verified from the experimental results reported elsewhere [32].

**Figure 2.** Distribution of electron density in the (111) crystal plane.

**Figure 3.** Simulation results of first principal stresses and shear stress: first principal stress in (**a1**) the TLP bonded joint and (**a2**) the intermediate layer; and shear stress in (**b1**) the TLP bonded joint and (**b2**) the intermediate layer.

The maximum shear stress theory assumes that failure occurs when the maximum shear stress reaches the yield point measured in the simple tension test, which is often used during the strength design of ductile materials. Because the maximum principal stress theory mentioned above is not always suitable for ductile materials, the maximum shear stress was used to evaluate the failure behavior of the TLP bonded joint. The maximum shear stress in each element can be expressed as

$$
\tau\_{\text{max}} = \frac{\sigma\_1 - \sigma\_3}{2},
\tag{5}
$$

where σ<sup>3</sup> is the third principal stress. Figure 3b1,b2 show the distribution of the maximum shear stresses in the submodel of a TLP bonded joint, where the maximum shear stress is located at the edge of the intermediate layer and at the interfaces between the intermediate layer and the parent alloy (refer to the red contour in the figure). In the intermediate layer, the maximum and average τmax are 179.9 MPa and 155.9 MPa respectively, and the ratio between the maximum τmax and the average τmax is 1.15. In addition, our calculation shows that if the intermediate layer is treated as a single alloy with the same orientation as the parent alloy, no stress concentration occurs in the joint and the τmax is 100 MPa.

To consider the influence of the second principal stress σ<sup>2</sup> on the stress concentration of a complex stress system, the von Mises stress (equivalent stress, σeq) was also calculated in this study. The von Mises hypothesis (i.e., the fourth strength theory) [33] holds that a material begins to yield when σeq reaches the yield strength and that σeq is given by Equation (6):

$$
\sigma\_{\rm eq} = \sqrt{\frac{(\sigma\_1 - \sigma\_2)^2 + (\sigma\_2 - \sigma\_3)^2 + (\sigma\_3 - \sigma\_1)^2}{2}}.\tag{6}
$$

Meanwhile, to evaluate the stress triaxiality, which affects the initiation and growth of micro-voids and cracks [34] on the failure of the TLP bonded joint, the damage equivalent stress (σ∗ eq) in the joint was also investigated. It can be written as follows [34]:

$$
\sigma\_{\rm eq}^{\*} = \sigma\_{\rm eq} \left[ \frac{2}{3} (1 + \nu) + 3(1 - 2\nu)(T\_{\sigma})^2 \right]^{\frac{1}{2}},\tag{7}
$$

where *T*σ is stress triaxiality, and

$$T\_{\sigma} = \sigma\_{\mathbf{m}} / \sigma\_{\mathbf{eq}}\tag{8}$$

$$
\sigma\_{\mathbf{m}} = (\sigma\_1 + \sigma\_2 + \sigma\_3) / 3,\tag{9}
$$

where σ<sup>m</sup> is the hydrostatic stress. Figure 4 shows the simulation results for σeq and σ<sup>∗</sup> eq in the TLP bonded joint. Obviously, both σeq and σ<sup>∗</sup> eq in the intermediate layer are greater than those in the parent alloy. The σeq and σ<sup>∗</sup> eq maximally distribute at the edges of the intermediate layer and at the interfaces between the intermediate layer and parent alloy (refer to the red contour in the figure), the locations of which are similar to those of the maximum τmax. The maximum σeq and σ<sup>∗</sup> eq in the intermediate layer are 350.7 GPa and 338.0 GPa respectively. If the intermediate layer is treated as a single alloy with the same orientation as the parent alloy, then no stress concentration occurs and the values of σeq and σ<sup>∗</sup> eq are 200 MPa (see Table S1).

**Figure 4.** Simulation results of the von Mises equivalent stress and the damage equivalent stress: von Mises equivalent stress in the (**a1**) TLP bonded joint and (**a2**) intermediate layer, and damage equivalent stress in the (**b1**) TLP bonded joint and (**b2**) intermediate layer.

Elastic modulus is the principal mechanical parameter for structural materials, which can be adjusted and controlled by adding alloyed elements or performing heat treatments. Figure 5 presents the influence of the elastic modulus of the intermediate layer (*E*inter) on the stress concentration, wherein the elastic modulus ranges from 0.85*E* to 1.15*E* (*E* = 207.378 GPa). Clearly, with the decreasing elastic modulus of the intermediate layer, the maximum stresses of first principal stress, maximum shear stress, equivalent stress, and damages equivalent stress decrease linearly. The ratio between the maximum stress and the average stress is also reduced, indicating that reducing *E*inter can relieve the stress concentration. Also, the average stresses decrease linearly with decreasing elastic modulus of the intermediate layer (see Supplementary Figure S2). Therefore, in addition to increasing the mechanical strength of the intermediate layer, the simulation result in this study suggests that appropriately reducing the elastic modulus of the intermediate layer can also improve the mechanical strength of a TLP bonded joint. To verify this opinion, additional experimental work was also carried out.

**Figure 5.** Influence of elastic moduli on (**a**) the maximum stresses and (**b**) the ratio between the maximum stress and the average stress in the TLP bonded joint.

In the experiment, single-crystalline Ni3Al-based superalloy (named IC10 alloy) samples were prepared by TLP bonding process in a high vacuum diffusion furnace (Centorr Vacuum Industries, Workhorse II, Nashua, NH, USA) and the bonding temperature, time, and pressure were 1250 ◦C, 6 h, and 5 MPa respectively. The samples of TLP bonded joints and their geometry have been given in Figure 1b,c above, and the grain boundary and crystal orientation in a typical zone in the joint are presented in Supplementary Figure S3. By nanoindentation testing, as illustrated in Supplementary Figure S4, the elastic modulus and hardness of the parent alloy and intermediate-layer alloy were measured. Test results show that the average elastic modulus of parent alloy (*E*parent) and the average elastic modulus of intermediate-layer alloy *E*inter are 221.4 GPa and 208.5 GPa respectively (see Supplementary Figure S5), and the ratio of *E*inter to *E*parent (i.e., *E*inter/*E*parent) is 0.942. Then, to reduce *E*inter and *E*inter/*E*parent, post weld heat treatment (PWHT) was carried out in a heat treatment furnace (SG-QF1400, Shanghai, China). During PWHT, samples of TLP bonded joints were heated to about 1200◦C and held for 6 h, and then cooled down in air to room temperature. Because that Boron was added to the intermediate-layer alloy as a melting-point depressant, the atomic percent of Boron in the intermediate-layer alloy was much higher than that in the parent alloy before PWHT. In the PWHT, Boron atoms diffused from the intermediate layer to the parent alloy. As the result of solution strengthening, after PWHT, *E*parent increased to 223.7 GPa while *E*inter decreased to 205.4 GPa, and *E*inter/*E*parent was reduced to 0.918. According to the result of tensile testing, the average tensile strength after PWHT is 819.0 MPa at room temperature, which is higher than before PWHT (i.e., 798.0 MPa) as shown in Supplementary Figure S6. Clearly, the decrease of *E*inter or *E*inter/*E*parent can improve the mechanical strength of a TLP bonded joint, which supports the simulation result in the manuscript. Notably, the ideal strength of the TLP bonded joint should be the same as or practically be as close as possible to the strength of the parent alloy. Usually, achieving a mechanical strength of a TLP bonded joint greater than 90% of the mechanical strength of a single crystal is difficult in experiments. To further improve the mechanical strength of TLP bonded joints, sufficient attention should be devoted to the stress concentration caused by the different states of crystal orientation between the parent alloy and intermediate layer.
