Single twinborn system dominated twinborn transmission probability prediction method

Abstract

The invention belongs to the technical field of material science foundation, and discloses a single twinborn system dominated twinborn transmission probability prediction method. According to the method, aiming at the condition that a single twin system is started preferentially under specific texture and force field conditions, the uniqueness of the twin system is limited and activated, and a three-dimensional matrix expression system of specific twin system spatial orientation is constructed. And based on a judgment rule that twin crystal faces and twin directions of the two crystal grains are parallel, calculating the minimum orientation difference capable of realizing parallel twin systems among the crystal grains. According to the method, the twinborn orientation difference is provided as a key physical quantity for quantifying the twinborn transmission capability, and the regulation and control mechanism of single twinborn system activation on local stress redistribution is disclosed by eliminating the interference of a non-activated twinborn system. The method is applicable to a twin-dominated deformation system at low temperature or high strain rate, and provides a micromechanical basis for optimizing strain coordination at the grain boundary.

Description

Technical Field

[0001] This invention belongs to the field of basic materials science and relates to a method for predicting twin transfer probability dominated by a single twin system. Background Technology

[0002] Twin deformation, as a key mechanism for plastic coordination in metallic materials, directly affects the synergy of dislocation movement at grain boundaries and the regulation of crack propagation paths. When dislocation slip accumulates at grain boundaries, stress concentration occurs. Activating twin deformation can absorb grain boundary dislocations on the one hand, and reconstruct the local stress state on the other hand to trigger slip of adjacent grains, ultimately achieving the goal of delaying macroscopic damage.

[0003] Under specific crystallographic conditions (such as low temperature, high strain rate, or strong texture), polycrystalline materials of the same crystal system may preferentially activate specific twin systems. For example, in room temperature compressive deformation under loading along the basal plane normal, {10-12} tensile twinning is often the dominant mechanism for plastic deformation in close-packed hexagonal metals. This selective activation only requires focusing on the spatial orientation of a single twin system, avoiding interference from other twin systems, to determine the twinning transfer capability across grain boundaries, thus improving computational efficiency.

[0004] Existing twin activation models mostly rely on the projection relationship between the twin plane normal and the twinning direction (such as the geometric compatibility factor). However, these methods do not consider the inherent constraints of crystal symmetry on the matching degree of the twin system, making it difficult to quantify the impact of the twin orientation deviation between the two grains on the activation probability. Therefore, constructing an activation prediction model based on a specific single twin orientation relationship is of great significance for resolving the deformation coordination mechanism at grain boundaries and optimizing the material strengthening and toughening design, while reducing computational complexity. Summary of the Invention

[0005] To address the aforementioned technical issues, a twin transmission probability prediction method dominated by a single twin system is proposed, the specific scheme of which is as follows:

[0006] A method for predicting twin transmission probability dominated by a single twin lineage includes the following steps:

[0007] Step 1: Obtain the crystal orientation information of the first and second grains and determine their crystal system;

[0008] Step 2: Determine the rotational symmetry operation matrix group of the crystal system, and obtain all variants and their quantities based on the determined single twin system;

[0009] Step 3: Construct a set of transformation matrices from crystal orientation to twin orientation, the number of which is the same as the number of twin variants;

[0010] Step 4: Based on the twin system spatial orientation relationship, calculate the minimum rotation angle required for the grain to rotate to form a specific parallel twin system with a second grain of the same crystal system;

[0011] The specific parallel twin system refers to the twin planes of two grains and the twinning directions on those planes being parallel, and the twin system is defined by a pre-determined single twin system;

[0012] The rotation angle is calculated using the rotation matrix of the grain, and the minimum value among all combinations is taken.

[0013] The minimum value obtained by calculation is defined as the twin orientation difference of a single twin system. The smaller the twin orientation difference, the greater the twin transmission probability.

[0014] The preferred embodiment of the single-twin-dominated twin transmission probability prediction method includes the following steps in its specific calculation process:

[0015] Step 1. Determine the two-dimensional crystal orientation matrices B1 and B2 of the two grains b1 and b2, with a shape of 3×3;

[0016] Determine the rotational symmetry operation matrix group of the crystal system to which the two grains belong, and superimpose the two-dimensional matrices in this group to form a three-dimensional matrix R. s , where s is the number of rotational symmetric operation matrices;

[0017] Determine the transformation matrix group from matrices b1 and b2 to known single twin orientations, with each twin variant corresponding to a transformation matrix. Superimpose all two-dimensional transformation matrices to form a three-dimensional matrix U. i , where i is the number of all twin variants in the twinning lineage;

[0018] In the formula, all variables with subscripts represent three-dimensional matrices, which are the result of superimposing two-dimensional matrices, similar to the superposition of RGB three-color layers in an image. Therefore, R s and U i The shapes are 3×3×s and 3×3×i, respectively;

[0019] Subscripts are used to refer to a three-dimensional matrix, and the subscript value is the number of stacked two-dimensional matrices in the third dimension. At the same time, operations on the three-dimensional matrix are performed sequentially on each stacked two-dimensional matrix.

[0020] Step 2. According to the definition of a parallel twin, we get:

[0021]

[0022] In the formula, D dIt is the rotation matrix of all orientations that can produce a parallel twin system when grain b1 is rotated to the position of grain b2, and the parallel twin system is determined by the given twin system. In addition, d = s×i×s×i.

[0023] Step 3. Rotation matrix D d Represented as:

[0024]

[0025] In the formula, U i and(U i ) -1 All are three-dimensional matrices. During the operation, the operation is performed on each two-dimensional matrix in the three-dimensional matrix in turn. Generally, U i and(U i ) -1 The two obtained two-dimensional matrices are not inverses of each other, so U cannot be used. i (U i ) -1 It can be directly simplified to the identity matrix E;

[0026] Step 4. Rotate by angle θ d The following relationship must be satisfied:

[0027]

[0028] In the formula, tr(D) d ) refers to D d Find the trace of each two-dimensional matrix stacked in the middle, and θ d The minimum value in the stacked matrix is ​​the twin orientation difference of a single twin system.

[0029] The preferred embodiment of the single twin-system-dominated twin transmission probability prediction method is that when the twin orientation difference is less than a threshold, it is determined that twin transmission is likely to occur; the threshold is set according to the material and deformation conditions, and is generally 12°.

[0030] Beneficial effects:

[0031] This method achieves crystallographic-driven prediction of twin propagation probability by constructing a quantitative index of the orientation matching degree between two specific twin systems of grains. Compared with traditional models that rely solely on the twin plane normal and twin direction projection, this method has significant advantages in the following aspects: First, by introducing the simplified characteristics of the matching relationship between the twin plane and twin direction through crystal symmetry, the rotation angle is used as a quantitative characterization of the twin system matching degree, making the calculation of activation probability more consistent with crystallographic principles; second, by limiting the calculation range of specific twin systems, the computational complexity of polycrystalline systems is effectively reduced, improving the prediction efficiency of deformation compatibility behavior at grain boundaries. This model is compatible with typical twin systems of various crystal systems, providing a computable theoretical tool for analyzing strain compatibility mechanisms at grain boundaries and optimizing the strength and toughness design of materials. Attached Figure Description

[0032] Figure 1 This is a diagram showing the grain and twin distribution and orientation of the titanium alloy in Example 1;

[0033] Figure 2 This is a diagram showing the grain and twin distribution and orientation of the titanium alloy in Example 2;

[0034] Figure 3 This is a diagram showing the grain and twin distribution and orientation of the titanium alloy in Example 3. Detailed Implementation

[0035] The following is in conjunction with the appendix Figure 1-3 The following is a detailed description of specific embodiments of the present invention; the following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0036] Example 1

[0037] This embodiment uses titanium alloy as the research object and employs the twin propagation probability prediction method dominated by the single twin system of this invention for calculation. In this example, the observed twins are mainly {11-21}<-1-126> tensile twins. Therefore, the twin system initiated by the grains is limited to {11-21}<-1-126> to improve computational efficiency.

[0038] Step 1. Based on the Euler angles of each group of grains in Table 1, determine the two-dimensional crystal orientation matrices B1 and B2 of the two grains b1 and b2 in each group of grains, with a shape of 3×3;

[0039] Table 1 Figure 1 Euler angles of each grain

[0040]

[0041] The crystal system of each grain in Table 1 is hexagonal close-packed crystal system. The corresponding rotational symmetry operation matrix groups are superimposed to form a three-dimensional matrix R. s, where s is the number of rotational symmetric operation matrices, as shown in Table 2;

[0042] Table 2. Rotational symmetry operation matrices for the hexagonal close-packed crystal system

[0043]

[0044] Determine the transformation matrix group from the matrices b1 and b2 to the known single twin orientation, with each twin variant corresponding to a transformation matrix. Superimpose all the two-dimensional transformation matrices to form a three-dimensional matrix U. i i is equivalent to the number of all twin variants in the twinning lineage, as shown in Table 3.

[0045] In the formula, all variables with subscripts represent three-dimensional matrices, which are the result of superimposing two-dimensional matrices, similar to the superposition of RGB three-color layers in an image. Therefore, R s and U i The shapes are 3×3×s and 3×3×i, respectively;

[0046] Subscripts are used to refer to a three-dimensional matrix, and the subscript value is the number of stacked two-dimensional matrices in the third dimension. At the same time, operations on the three-dimensional matrix are performed sequentially on each stacked two-dimensional matrix.

[0047] Table 3 Orientation transformation matrix of hexagonal close-packed grain twins

[0048]

[0049] Step 2. According to the definition of a parallel twin, we get:

[0050]

[0051] In the formula, D d It is the rotation matrix d = s×i×s×i, which rotates grain b1 to all parallel twin systems that can be generated based on a given twin system with grain b2.

[0052] Step 3. Rotation matrix D d Represented as:

[0053]

[0054] In the formula, U i (U i ) -1 It cannot be directly simplified to the identity matrix E because U i and(U i ) -1 Both are three-dimensional matrices, and operations on three-dimensional matrices refer to performing operations on each of their two-dimensional matrices sequentially. Generally, U...i and(U i ) -1 The two two-dimensional matrices obtained are not inverses of each other.

[0055] Step 4. Rotate by angle θ d The following relationship must be satisfied:

[0056]

[0057] In the formula, tr(D) d ) refers to D d Find the trace of each two-dimensional matrix stacked in the middle, and θ d The minimum value in the stacked matrix is ​​the twin orientation difference of a single twin system.

[0058] Table 4 Figure 1 Twin orientation difference of each group of grains

[0059]

[0060] Results: Based on the grain orientations in Table 1, the corresponding twinning orientation differences can be calculated as shown in Table 4. The twinning orientation differences for all groups are greater than 12°, predicting no twinning propagation. This conclusion is consistent with... Figure 1 The experimental observations remained consistent.

[0061] Example 2

[0062] The twin propagation probability prediction method dominated by the single twin system of this invention is used for calculation. In this example, the observed twins are mainly {11-22}<11-2-3> compressed twins. Therefore, the twin system initiated by the grain is limited to {11-22}<11-2-3> to improve the calculation efficiency.

[0063] Step 1. Based on the Euler angles of each group of grains in Table 5, determine the two-dimensional crystal orientation matrices B1 and B2 of the two grains b1 and b2 in each group of grains, with a shape of 3×3;

[0064] Table 5 Figure 2 Euler angles of each grain

[0065]

[0066] The crystal system of each grain in Table 5 is hexagonal close-packed. The corresponding rotational symmetry operation matrix groups are superimposed to form a three-dimensional matrix R. s , where s is the number of rotational symmetric operation matrices, as shown in Table 2;

[0067] Determine the transformation matrix group from the matrices b1 and b2 to the known single twin orientation, with each twin variant corresponding to a transformation matrix. Superimpose all the two-dimensional transformation matrices to form a three-dimensional matrix U.i i is equivalent to the number of all twin variants in the twin lineage, as shown in Table 6.

[0068] In the formula, all variables with subscripts represent three-dimensional matrices, which are the result of superimposing two-dimensional matrices, similar to the superposition of RGB three-color layers in an image. Therefore, R s and U i The shapes are 3×3×s and 3×3×i, respectively;

[0069] Subscripts are used to refer to a three-dimensional matrix, and the subscript value is the number of stacked two-dimensional matrices in the third dimension. At the same time, operations on the three-dimensional matrix are performed sequentially on each stacked two-dimensional matrix.

[0070] Table 6 Orientation transformation matrix of hexagonal close-packed grain twins

[0071]

[0072] Step 2. According to the definition of a parallel twin, we get:

[0073]

[0074] In the formula, D d It is the rotation matrix d = s×i×s×i, which rotates grain b1 to all parallel twin systems that can be generated based on a given twin system with grain b2.

[0075] Step 3. Rotation matrix D d It can be represented as:

[0076]

[0077] In the formula, U i (U i ) -1 It cannot be directly simplified to the identity matrix E because U i and(U i ) -1 Both are three-dimensional matrices, and operations on three-dimensional matrices refer to performing operations on each of their two-dimensional matrices sequentially. Generally, U... i and(U i ) -1 The two two-dimensional matrices obtained are not inverses of each other.

[0078] Step 4. Rotate by angle θ d The following relationship must be satisfied:

[0079]

[0080] In the formula, tr(D)d ) refers to D d Find the trace of each two-dimensional matrix stacked in the middle, and θ d The minimum value in the stacked matrix is ​​the twin orientation difference of a single twin system.

[0081] Table 7 Figure 2 Twin orientation difference of each group of grains

[0082]

[0083] Results: Based on the grain orientations in Table 4, the corresponding twinning orientation differences were calculated as shown in Table 7. The twinning orientation differences for all groups were greater than 12°, indicating no twinning propagation was predicted. This conclusion is consistent with... Figure 2 The experimental observation results are consistent.

[0084] Example 3

[0085] The twin propagation probability prediction method dominated by the single twin system of this invention is used for calculation. In this example, the observed twins are mainly {10-12}<-1011> stretched twins. Therefore, the twin system initiated by the grain is limited to {10-12}<-1011> to improve the calculation efficiency.

[0086] Step 1. Based on the Euler angles of each group of grains in Table 8, determine the two-dimensional crystal orientation matrices B1 and B2 of the two grains b1 and b2 in each group of grains, with a shape of 3×3;

[0087] Table 8 Figure 3 Euler angles of each grain

[0088]

[0089] The crystal system of each grain in Table 6 is hexagonal close-packed. The corresponding rotational symmetry operation matrix groups are superimposed to form a three-dimensional matrix R. s , where s is the number of rotational symmetric operation matrices, as shown in Table 2;

[0090] Determine the transformation matrix group from the matrices b1 and b2 to the known single twin orientation, with each twin variant corresponding to a transformation matrix. Superimpose all the two-dimensional transformation matrices to form a three-dimensional matrix U. i i is equivalent to the number of all twin variants in the twinning lineage, as shown in Table 9.

[0091] In the formula, all variables with subscripts represent three-dimensional matrices, which are the result of superimposing two-dimensional matrices, similar to the superposition of RGB three-color layers in an image. Therefore, R s and U i The shapes are 3×3×s and 3×3×i, respectively;

[0092] Subscripts are used to refer to a three-dimensional matrix, and the subscript value is the number of stacked two-dimensional matrices in the third dimension. At the same time, operations on the three-dimensional matrix are performed sequentially on each stacked two-dimensional matrix.

[0093] Table 9 Orientation transformation matrix of hexagonal close-packed grain twins

[0094]

[0095] Step 2. According to the definition of a parallel twin, we get:

[0096]

[0097] In the formula, D d It is the rotation matrix d = s×i×s×i, which rotates grain b1 to all parallel twin systems that can be generated based on a given twin system with grain b2.

[0098] Step 3. Rotation matrix D d Represented as:

[0099]

[0100] In the formula, U i (U i ) -1 It cannot be directly simplified to the identity matrix E because U i and(U i ) -1 Both are three-dimensional matrices, and operations on three-dimensional matrices refer to performing operations on each of their two-dimensional matrices sequentially. Generally, U... i and(U i ) -1 The two two-dimensional matrices obtained are not inverses of each other.

[0101] Step 4. Rotate by angle θ d The following relationship must be satisfied:

[0102]

[0103] In the formula, tr(D) d ) refers to D d Find the trace of each two-dimensional matrix stacked in the middle, and θ d The minimum value in the stacking matrix represents the twin orientation difference of a single twin system. Based on the grain orientations in Table 4, the corresponding twin orientation differences can be calculated as shown in Table 10. Generally, two grains with a twin orientation difference of less than 12° are more likely to undergo twinning propagation. This conclusion is consistent with... Figure 3 The experimental observation results are consistent.

[0104] Table 10 Figure 3 Twin orientation difference of each group of grains

[0105]

[0106] Results: Based on the grain orientations in Table 10, the twinning orientation difference in group 1 is less than 12°, predicting twinning propagation; the difference in group 2 is greater than 12°, predicting no twinning propagation. This conclusion is consistent with... Figure 3 The experimental observation results are consistent.

[0107] The above embodiments are only for illustrating the technical concept and specific calculation method of the present invention, and are not intended to limit the ideas of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention, such as the selection of other alloy systems, should be included within the protection scope of the present invention.

[0108] The above description is only a preferred embodiment of the present invention and is not intended to limit the ideas of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for predicting twin transmission probability dominated by a single twin lineage, characterized in that, Includes the following steps: Step 1: Obtain the crystal orientation information of the first and second grains and determine their crystal system; Step 2: Determine the rotational symmetry operation matrix group of the crystal system, and obtain all variants and their quantities based on the determined single twin system; Step 3: Construct a set of transformation matrices from crystal orientation to twin orientation, the number of which is the same as the number of twin variants; Step 4: Calculate the minimum rotation angle required for the first grain to rotate to form a specific parallel twin system with the second grain using the twin system spatial orientation relationship; The specific parallel twin system refers to the twin planes of two grains and the twinning directions on those planes being parallel, and the twin system is defined by a pre-determined single twin system; The rotation angle is calculated using the rotation matrix of the grain, and the minimum value among all combinations is taken. The minimum value obtained by calculation is defined as the twin orientation difference of a single twin system. The smaller the twin orientation difference, the greater the twin transmission probability.

2. The twin transmission probability prediction method dominated by a single twin lineage according to claim 1, characterized in that, The specific calculation process includes the following steps: Step 1. Determine the two-dimensional crystal orientation matrices B1 and B2 of the two grains b1 and b2, with a shape of 3×3; Determine the rotational symmetry operation matrix group of the crystal system to which the two grains belong, and superimpose the two-dimensional matrices in this group into a three-dimensional matrix R. s , where s is the number of rotational symmetric operation matrices; Determine the transformation matrix group from matrices b1 and b2 to known single twin orientations, with each twin variant corresponding to a transformation matrix. Superimpose all two-dimensional transformation matrices to form a three-dimensional matrix U. i , where i is the number of all twin variants in the twinning lineage; In the formula, all variables with subscripts represent three-dimensional matrices, which are the result of superimposing two-dimensional matrices, similar to the superposition of RGB three-color layers in an image. Therefore, R s and U i The shapes are 3×3×s and 3×3×i, respectively; Subscripts are used to refer to a three-dimensional matrix, and the subscript value is the number of stacked two-dimensional matrices in the third dimension. At the same time, operations on the three-dimensional matrix are performed sequentially on each stacked two-dimensional matrix. Step 2. According to the definition of a parallel twin, we get: In the formula, D d It is the rotation matrix of all orientations that can produce a parallel twin system when grain b1 is rotated to the position of grain b2, and the parallel twin system is determined by the given twin system. In addition, d = s×i×s×i. Step 3. Rotation matrix D d Represented as: In the formula, U i and(U i ) -1 All are three-dimensional matrices. During the operation, the operation is performed on each two-dimensional matrix in the three-dimensional matrix in turn. Generally, U i and(U i ) -1 The two obtained two-dimensional matrices are not inverses of each other, so U cannot be directly used. i (U i ) -1 Simplified to the identity matrix E; Step 4. Rotate by angle θ d The following relationship must be satisfied: In the formula, tr(D) d ) refers to D d Find the trace of each two-dimensional matrix stacked in the middle, and θ d The minimum value in the stacked matrix is ​​the twin orientation difference of a single twin system.

3. The twin transmission probability prediction method dominated by a single twin lineage according to claim 1, characterized in that, When the twin orientation difference is less than a set threshold, it is determined that twin transmission is likely to occur; the threshold is set according to the material and deformation conditions, and is generally 12°.

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