Substrate with heat generator

Abstract

The present invention inhibits thermal cracking.　The present invention includes a substrate having a fracture toughness value of 1 [MPa·m1 / 2] or less and a heat generator provided to the substrate. When the Young's modulus of the substrate is expressed by E [MPa], the average coefficient of linear expansion of the substrate at 20-300°C is expressed by α [1 / °C], the square root of the area of a portion where the heat generator and the substrate lie on each other in a plan view is expressed by a [mm], the distance in the plan view from a highest-temperature point in the heat generator to an edge of the substrate is expressed by b [mm], and the difference between the temperature at the highest-temperature point and the ambient temperature is expressed by ΔT, then the conditional expressions c1×(c2+c3×a+c4×b+c5×ΔT)≤6.9 and c1=E×α / 0.208 are established, wherein when a is less than 50 mm, then c2=-2.279, c3=0.154, c4=-0.037, and c5=0.069; and when a is 50 mm or more, then c2=0.116, c3=0.054, c4=-0.044, and c5=0.101.

Description

Heating element-equipped substrate

[0001] The present disclosure relates to a substrate with a heating element.

[0002] BACKGROUND ART Conventionally, a glass plate provided with an antenna conductor has been known (see, for example, Patent Document 1 listed below).

[0003] International Publication No. 2020 / 095818

[0004] When a heating element such as an antenna conductor is mounted on a brittle substrate, the substrate may crack due to heat generated by the heating element, a phenomenon known as thermal cracking.

[0005] The present disclosure provides a substrate with a heating element that can suppress thermal cracking.

[0006] The first aspect is a fracture toughness value of 1 [MPa m １／２ ] or less, and a heating element provided on the substrate, wherein the Young's modulus of the substrate is E [MPa], the average linear expansion coefficient of the substrate at 20 to 300°C is α [1 / °C], the square root of the area where the heating element overlaps with the substrate in a plan view is a [mm], the distance in a plan view from the hottest point of the heating element to the edge of the substrate is b [mm], and the difference between the temperature of the hottest point and the ambient temperature is ΔT, the following conditional formula is established: c1 x (c2 + c3 x a + c4 x b + c5 x ΔT) ≦ 6.9 c1 = E x α / 0.208, when a is less than 50 mm, c2 = -2.279, c3 = 0.154, c4 = -0.037, c5 = 0.069, and when a is 50 mm or more, This is a substrate with a heating element, where c2=0.116, c3=0.054, c4=-0.044, and c5=0.101.

[0007] A second aspect is the substrate with a heating element of the first aspect, wherein the heating element may include a portion provided on a surface of the substrate.

[0008] A third aspect is the substrate with a heating element of the first or second aspect, wherein the heating element may include a conductor formed on the surface of the substrate.

[0009] A fourth aspect is the heater-equipped substrate of the third aspect, wherein the conductor may include a radiating element.

[0010] A fifth aspect is the substrate with a heater according to the third or fourth aspect, wherein the conductor may include a filter.

[0011] A sixth aspect is the substrate with a heating element according to any one of the first to fifth aspects, wherein the heating element may include a component that contacts the surface of the substrate.

[0012] A seventh aspect is the substrate with a heating element according to any one of the first to sixth aspects, wherein the thickness of the substrate is t [m], the thermal conductivity of the substrate is λ [W / (m·K)], and the overall heat transfer coefficient of the substrate is β [W / (m ２ ·K)], the following may be true: (t / λ)<(1 / β)×0.10.

[0013] An eighth aspect is the heater-equipped substrate according to any one of the first to seventh aspects, wherein the thickness of the substrate may be 2 mm or more and 19 mm or less.

[0014] A ninth aspect is the heater-equipped substrate according to any one of the first to eighth aspects, wherein the substrate may include a glass plate.

[0015] A tenth aspect is the substrate with a heating element of any one of the first to ninth aspects, wherein the substrate may include a plurality of layers having different Young's moduli or average linear expansion coefficients, and c1 in the conditional formula may be the largest value among the c1 of each of the plurality of layers.

[0016] An eleventh aspect is the heater-equipped substrate according to any one of the first to tenth aspects, wherein the substrate may be a glass plate for use in buildings.

[0017] A twelfth aspect is the heater-equipped substrate according to any one of the first to tenth aspects, wherein the substrate may be a glass plate for a vehicle.

[0018] According to the present disclosure, it is possible to provide a substrate with a heating element that can suppress thermal cracking.

[0019] FIG. 1 is a cross-sectional view of an example of a substrate with a heater according to an embodiment; FIG. 2 is a plan view of an example of a substrate with a heater according to an embodiment; FIG. 3 is a diagram showing calculation conditions and calculation results of the finite element method used to derive condition A; FIG. 4 is a diagram showing calculation conditions and calculation results of the finite element method used to derive condition A; FIG. 5 is a diagram for explaining the prediction accuracy using a regression equation obtained by multiple regression analysis; FIG. 6 is a table comparing the temperatures on both surfaces of a substrate (glass plate) having a thermal conductivity λ of 1 [W / (m·K)] and a thickness t of 2 mm or more and 19 mm or less; FIG. 7 is a table illustrating the layer structure of a substrate and each physical property value; FIG. 8 is a diagram for explaining the prediction accuracy using a regression equation obtained by multiple regression analysis;

[0020] Hereinafter, embodiments will be described with reference to the drawings. To facilitate understanding, the scale of each component in the drawings may differ from the actual scale. Directions such as parallel, right-angled, orthogonal, horizontal, vertical, up, down, left, and right are permitted to deviate to the extent that they do not impair the functions and effects of the embodiments. The shape of the corners is not limited to right angles and may be rounded in an arched shape. Parallel, right-angled, orthogonal, horizontal, and vertical may include approximately parallel, approximately right-angled, approximately orthogonal, approximately horizontal, and approximately vertical.

[0021] In this specification, a three-dimensional Cartesian coordinate system with three axes (X-axis, Y-axis, and Z-axis) is used, where the width or lateral direction of the substrate is defined as the X-axis, the height or longitudinal direction of the substrate is defined as the Y-axis, and the thickness direction of the substrate is defined as the Z-axis. The direction from bottom to top of the substrate is defined as the +Y-axis, and the opposite direction is defined as the −Y-axis.

[0022] The X-axis, Y-axis, and Z-axis directions represent directions parallel to the X-axis, Y-axis, and Z-axis, respectively. The X-axis, Y-axis, and Z-axis directions are perpendicular to one another. The XY plane, YZ plane, and ZX plane represent imaginary planes parallel to the X-axis and Y-axis directions, imaginary planes parallel to the Y-axis and Z-axis directions, and imaginary planes parallel to the Z-axis and X-axis directions, respectively.

[0023] For example, the X-axis direction and the Z-axis direction are substantially parallel to a direction parallel to a horizontal plane (horizontal direction), and the Y-axis direction is substantially parallel to a vertical direction perpendicular to the horizontal plane.

[0024] Fig. 1 is a cross-sectional view of an example of a substrate with a heater element according to an embodiment. The substrate with a heater element shown in Fig. 1 includes a substrate 50 and a heater element 40. The substrate 50 and the heater element 40 are stacked in the Z-axis direction.

[0025] The substrate 50 has a surface 51 facing the −Z-axis direction and a surface 52 facing the +Z-axis direction. The thickness t of the substrate 50 in the Z-axis direction corresponds to the distance between the surface 51 and the surface 52.

[0026] The substrate 50 has a fracture toughness of 1 [MPa m １／２ The fracture toughness value corresponds to the critical value of the stress intensity factor when fracture occurs. １／２ ] Examples of the following materials include silica glass, soda lime glass, borosilicate glass, alkali-free glass, and aluminosilicate glass.

[0027] The substrate 50 is not limited to a single plate, and may be laminated glass, double-glazed glass, Low-E glass, light-control glass, or glass containing wire-shaped elements. Low-E glass is also called low-emissivity glass, and may be window glass coated on one surface with a coating layer (transparent conductive film) that has heat-reflecting properties. The glass containing wire-shaped elements may be wired glass sheet in which a metal mesh is inserted inside the glass, or wired glass sheet in which metal wires that are not wire-shaped are inserted inside the glass.

[0028] The substrate 50 may be a substrate used for a display, a solar panel, an antenna, a window glass, etc. The substrate 50 may be a glass plate for a building or a vehicle. Examples of glass plates for a building include window glass, glass facades, and interior glass. Examples of glass plates for a vehicle include window glass and interior glass. The uses of the substrate 50 with the heating element 40 are not limited to these.

[0029] The heating element 40 generates heat by itself, and generates heat, for example, when an electric current flows through the heating element 40 to cause the heating element 40 to perform its function.

[0030] The heating element 40 is provided on the substrate 50. The heating element 40 is provided on at least one of the surface and the interior of the substrate 50. Fig. 1 illustrates a case where the heating element 40 includes a portion provided on the surface 51 of the substrate 50. The heating element 40 may also include a portion provided on the surface 52. When the substrate 50 includes multiple dielectric layers, the heating element 40 may be located between the multiple dielectric layers included in the substrate 50.

[0031] When the heating element 40 includes a portion provided on the surface of the substrate 50, the portion may be a conductor formed by printing or the like on the surface 51 or the surface 52 of the substrate 50. Examples of such conductors include conductors for antennas (radiating elements, filters, feeding elements, etc.), electrical wiring, heater wiring, etc.

[0032] When the heating element 40 includes a portion provided on the surface of the substrate 50, the heating element 40 may include a component that contacts the surface 51 or the surface 52 of the substrate 50. Examples of such components include electronic components, electrical components, and mechanical components. Examples of electronic components include transistors and integrated circuits (ICs). Examples of ICs include analog ICs, driver ICs, power supply ICs, communication ICs, processors, memories, microcomputers, field programmable gate arrays (FPGAs), application specific integrated circuits (ASICs), and large scale integrations (LSIs). The heating element 40 may include a resistive element through which a current flows.

[0033] The manner in which the heating element 40 comes into contact with the surface of the substrate 50 is not limited to a manner in which the heating element 40 comes into direct contact with the surface of the substrate 50, but may also include a manner in which the heating element 40 comes into indirect contact with the surface via a thermally conductive member (for example, solder, adhesive, etc.). For example, the heating element 40 may be attached to a circuit formed on the surface of the substrate 50 by soldering or adhesive, or may be attached by fitting or fastening into a hole formed in the substrate 50.

[0034] Fig. 2 is a plan view of an example of a substrate with a heating element according to one embodiment. The shapes of the substrate 50 and the heating element 40 are not particularly limited and may be rectangular or circular. Fig. 2 illustrates a case where the shapes of the substrate 50 and the heating element 40 in plan view are each rectangular.

[0035] The heating element 40 shown in FIG. 2 has a square shape with dimensions a × a in plan view. The distance b in plan view from the highest temperature point 41 generated by the heating element 40 to the edge 53 of the substrate 50 is defined as the distance from the highest temperature point 41 to the closest edge of the peripheral edge of the substrate 50. In a configuration in which the heating element 40 is provided on the substrate 50 as shown in FIG. 2 , the larger the dimension a or the shorter the distance b, the greater the stress generated at the edge 53 closest to the highest temperature point 41, increasing the likelihood of thermal cracking of the substrate 50. Therefore, the present disclosure provides a substrate with a heating element that can suppress thermal cracking of the substrate 50. Depending on the conditions, the maximum stress generated in the substrate plane may occur within the plane (e.g., near the heating element). However, since brittle materials with low fracture toughness values, which are the subject of this disclosure, generally have lower fracture stress at the edge than within the plane, we will focus only on the stress generated at the edge. The maximum stress occurs at the edge of the substrate 50 at the shortest distance from the highest temperature point 41 in plan view.

[0036] 2 , the stress generated in the substrate 50 is σ [MPa], the Young's modulus of the substrate 50 is E [MPa], the average linear expansion coefficient of the substrate 50 at 20 to 300° C. is α [1 / ° C.], the square root of the area where the heating element 40 overlaps with the substrate 50 in a planar view is a [mm], the distance from the hottest point 41 of the heating element 40 to the edge 53 of the substrate 50 in a planar view is b [mm], and the difference in temperature between the hottest point 41 and the ambient temperature is ΔT. The square root of the area where the heating element 40 overlaps with the substrate 50 in a planar view (=dimension a) represents the average size of the heating element 40. The temperature of the hottest point 41 can be measured by thermography.

[0037] The substrate 50 with heating element 40 satisfies the following conditional formula A1: c1 x (c2 + c3 x a + c4 x b + c5 x ΔT) ≦ 6.9 (formula 1) c1 = E x α / 0.208 (formula 2). When a is less than 50 mm, c2 = -2.279, c3 = 0.154, c4 = -0.037, c5 = 0.069. When a is 50 mm or more, c2 = 0.116, c3 = 0.054, c4 = -0.044, c5 = 0.101 (condition A).

[0038] When the substrate 50 with the heating element 40 satisfies the condition A, thermal cracking of the substrate 50 is suppressed, as will be shown in the simulation and judgment examples described later.

[0039] The left side of Equation 1 corresponds to the stress σ [MPa] generated at the edge 53 of the substrate 50. By selecting the dimension a, distance b, and temperature difference ΔT so that the value of the left side of Equation 1 calculated from Young's modulus E, average linear expansion coefficient α, dimension a, distance b, and temperature difference ΔT is equal to or less than 6.9 [MPa], which is the edge fracture allowable stress of the substrate 50, thermal cracking of the substrate 50 can be suppressed. The value 6.9 on the right side of Equation 1 is approximately equal to the value of the long-term edge fracture allowable stress [MPa] of soda-lime glass with a thickness t of 20 mm or less. The fracture toughness value is 1 [MPa m １／２ ] or less, the edge fracture allowable stress of the substrate is not thought to vary significantly depending on the substrate material.

[0040] Equation 1 is a regression equation similar to Hooke's law. c1 corresponds to the Young's modulus of borosilicate glass and the Young's modulus normalized by the average linear expansion coefficient. (c2 + c3 × a + c4 × b + c5 × ΔT) corresponds to the strain of the substrate 50. c2, c3, c4, and c5 are regression coefficients. "0.208" in Equation 2 corresponds to "6.4 × 10 ４ x 3.25 x 10 －６ " is the calculated value. "6.4 x 10 ４ " corresponds to the Young's modulus [MPa] of borosilicate glass. "3.25 × 10 －６ " corresponds to the average linear expansion coefficient [1 / °C] of borosilicate glass at 20 to 300°C.

[0041] Next, a method for deriving the condition A through simulation will be described.

[0042] 3 and 4 are diagrams showing the calculation conditions and calculation results of the finite element method (FEM) used to derive condition A. Under the condition that the heating element 40 is placed on the surface 51 of the substrate 50, the dimension a, distance b, and the amount of heat generated (temperature) of the heating element 40 are changed, and the temperature at the hottest point 41 and the distribution of stress σ at the edge 53 (right half of FIG. 2) are evaluated. Two types of square borosilicate glass substrates 50 with sides of 400 mm and 1000 mm are assumed, depending on the dimensions of the heating element 40. Other conditions during the evaluation are: Ambient temperature Tair: 25°C; Thickness t of the substrate 50: 5 mm; Overall heat transfer coefficient of the surface 51 of the substrate 50: 4 W / m ２ ·K] (assuming a quiet room) Young's modulus E of substrate 50: 6.4 × 10 ４ [MPa] Average linear expansion coefficient α of the substrate 50: 3.25 × 10 －６ [1 / ° C.] Thermal conductivity of the substrate 50: 1.2 [W / m·K] Poisson's ratio of the substrate 50: 0.2.

[0043] Fig. 5 is a diagram for explaining the estimation accuracy of the regression equation obtained by multiple regression analysis. The horizontal axis represents the maximum value of stress σ calculated by the FEM shown in Fig. 3 and Fig. 4. The vertical axis represents the value of the left side of Equation 1 calculated by the regression equation (Condition A) derived by multiple regression analysis using the least squares method from the FEM calculation conditions and calculation results shown in Fig. 3 and Fig. 4, and represents the stress σ.

[0044] When a is 2 mm or more and less than 50 mm, the coefficient of determination R of the regression equation (condition A) when "c2 = -2.279, c3 = 0.154, c4 = -0.037, c5 = 0.069" is used. ２ When a is 50 mm or more and 200 mm or less, the coefficient of determination R of the regression equation (condition A) when "c2 = 0.116, c3 = 0.054, c4 = -0.044, c5 = 0.101" is ２ is 0.89. Thus, the coefficient of determination R ２ is relatively high, it can be said that the regression equation (condition A) has high accuracy in estimating the stress σ.

[0045] Next, the thickness of the substrate 50 will be described.

[0046] The thickness of the substrate 50 is t [m], the thermal conductivity of the substrate 50 is λ [W / (m·K)], and the overall heat transfer coefficient of the substrate 50 is β [W / (m ２ ·K)]. In this case, in order to suppress thermal cracking of the substrate 50, it is preferable to set the thickness t to a length that satisfies the following expression: (t / λ)<(1 / β)×0.10 ...Equation 3a. In order to suppress thermal cracking of the substrate 50, it is more preferable to set the thickness t to a length that satisfies the following expression: (t / λ)<(1 / β)×0.05 ...Equation 3b, and it is even more preferable to set the thickness t to a length that satisfies the following expression: (t / λ)<(1 / β)×0.02 ...Equation 3c.

[0047] Alternatively, in order to prevent thermal cracking of the substrate 50, it is preferable to set the thickness t to a length of 2 mm or more and 19 mm or less.

[0048] 1, assume that heat from the heating element 40 is transferred from the surface 51 to the surface 52 through the substrate 50. If the thickness of the substrate 50 is t [m] and the thermal conductivity of the substrate 50 is λ [W / (m·K)], then the thermal resistance R1 [m ２ When the substrate 50 is a glass plate, if the representative value is λ=1, and the thickness t is 2 mm or more and 19 mm or less, the thermal resistance R1 is 0.002 [m ２ ・K / W] or more 0.019 [m ２ On the other hand, the overall heat transfer coefficient of the surface 51 is β [W / (m ２ ·K)], the thermal resistance R2 [m ２ When the substrate 50 is a glass plate, if the representative value is β=4 (a representative value in the case of calm air), the thermal resistance R2 is 0.25 [m ２ ・K / W].

[0049] Assuming that substrate 50 is a glass plate with a thickness t of 2 mm or more and 19 mm or less, the thermal resistance R1 in the thickness direction of substrate 50 is sufficiently smaller than the thermal resistance R2 of surface 51 of substrate 50 (R1 < R2 × 0.1; in other words, equation 3a holds). Therefore, temperature T1 of surface 51 is approximately equal to temperature T2 of surface 52. Therefore, if thickness t is 2 mm or more and 19 mm or less, the temperature distribution in the thickness direction of substrate 50 is small, and the change in stress σ due to differences in thickness t is small.

[0050] 6 is a table comparing the temperature T1 of surface 51 and the temperature T2 of surface 52 of substrate 50 (glass plate) having a thermal conductivity λ of 1 [W / (m·K)] and a thickness t of 2 mm to 19 mm. As illustrated in FIG. 6, regardless of the thickness t, the temperature T1 of surface 51 is substantially equal to the temperature T2 of surface 52.

[0051] Therefore, in order to suppress thermal cracking of the substrate 50, it is preferable that the above formula 3a, 3b or 3c is satisfied, or the thickness t is 2 mm or more and 19 mm or less.

[0052] Furthermore, if the thickness t is 2 mm or more, sufficient strength is obtained for attaching the three-dimensional heating element 40 to the surface 51. If the thickness t is 19 mm or less, the radio wave transmission performance of the substrate 50 is good when the heating element 40 includes an antenna. The thickness t of the substrate 50 is more preferably 3 mm or more and 15 mm or less, and even more preferably 6 mm or more and 12 mm or less.

[0053] Next, a description will be given of the layer structure of the substrate 50. The layer structure of the substrate 50 is not limited to a single layer, but may be a multi-layer structure. The substrate 50 may include a dielectric layer such as a glass plate.

[0054] The substrate 50 may include multiple layers having different Young's moduli E or average linear expansion coefficients α. In order to suppress thermal cracking of the substrate 50, it is preferable that c1 in the above conditional formula A1 be set to the largest value among the c1 values ​​of the multiple layers. This makes it possible to suppress thermal cracking of the layer that is most susceptible to cracking among the multiple layers.

[0055] 7 is a table illustrating the layer configuration of the substrate and the physical properties of each layer. The substrate 50 includes, in order from the first layer to the fifth layer, borosilicate glass, OCA (Optical Clear Adhesive), borosilicate glass, OCA, and borosilicate glass. In this case, the value of c1 in the above-described conditional formula A1, 1.2, which is the largest value of c1 among the multiple layers, is used.

[0056] FIG. 8 is a diagram illustrating the estimation accuracy of the regression equation obtained by multiple regression analysis. FIG. 8 shows the results of evaluating the stress σ generated at the edge 53 when the temperature difference ΔT was varied from 50°C to 125°C for a substrate 50 including multiple layers (shown in FIG. 7 ), in which two heating elements 40 measuring 20 mm in length and 30 mm in width were located between the first and third borosilicate glass layers and bonded together with an OCA. The distance b for one heating element 40 was 60 mm, and the distance b for the other heating element 40 was 85 mm. The vertical axis represents the maximum stress σ calculated by FEM. The horizontal axis represents the value of the left side of Equation 1, calculated using the regression equation (Condition A) derived by multiple regression analysis using the least squares method from the FEM calculation conditions and calculation results shown in FIGS. 3 and 4 , and represents the stress σ. As shown in FIG. 8 , both when the distance b was 60 mm and when it was 85 mm, the stress σ was plotted along a straight line with a slope of 1. Therefore, even if the substrate 50 includes multiple layers or the heating element 40 includes multiple heating elements, it can be said that the estimation accuracy of the stress σ obtained from the regression equation (condition A) is high.

[0057] Fig. 9 is a table showing an example of determination based on condition A. In Fig. 9, for a plurality of types of substrates 50 with heating elements 40 that differ in any of dimension a, distance b, temperature difference ΔT, and coefficient c1, cases where condition A is satisfied, that is, the value of the left side of equation 1 calculated from conditional equation A1 is 6.9 or less, are indicated as "OK," and cases where condition A is not satisfied are indicated as "NG." By selecting a case where condition A is satisfied as the parameters of the substrate 50 with heating elements 40, thermal cracking of the substrate 50 can be suppressed.

[0058] Although the embodiments have been described above, the above embodiments are presented as examples, and the present invention is not limited to the above embodiments. The above embodiments can be implemented in various other forms, and various combinations, omissions, substitutions, modifications, etc. are possible within the scope of the invention. These embodiments and their variations are included within the scope and summary of the invention, as well as within the scope of the inventions described in the claims and their equivalents. The entire contents of the specification, claims, drawings, and abstract of Japanese Patent Application No. 2023-148377, filed on September 13, 2023, are hereby incorporated by reference as the disclosure of the specification of the present invention.

[0059] 40 Heating element 41 Highest temperature point 50 Substrate 51, 52 Surface 53 Edge

Claims

1. Fracture toughness value is 1 [MPa m １／２ ] or less, and a heating element provided on the substrate, wherein the Young's modulus of the substrate is E [MPa], the average linear expansion coefficient of the substrate at 20 to 300°C is α [1 / °C], the square root of the area of ​​the heating element overlapping with the substrate in a planar view is a [mm], the distance in a planar view from the hottest point of the heating element to the edge of the substrate is b [mm], and the difference between the temperature of the hottest point and the atmospheric temperature is ΔT, the following conditional formula is established: c1 x (c2 + c3 x a + c4 x b + c5 x ΔT) ≦ 6.9 c1 = E x α / 0.208, when a is less than 50 mm, c2 = -2.279, c3 = 0.154, c4 = -0.037, c5 = 0.069, and when a is 50 mm or more, A substrate with a heating element, where c2 = 0.116, c3 = 0.054, c4 = -0.044, and c5 = 0.

101.

2. The substrate with a heater according to claim 1, wherein the heater includes a portion provided on the surface of the substrate.

3. The substrate with a heating element according to claim 2, wherein the heating element includes a conductor formed on the surface of the substrate.

4. The substrate with a heating element according to claim 3, wherein the conductor includes a radiating element.

5. The substrate with a heating element according to claim 3, wherein the conductor includes a filter.

6. The substrate with a heating element according to claim 2, wherein the heating element includes a component that contacts the surface of the substrate.

7. The thickness of the substrate is t [m], the thermal conductivity of the substrate is λ [W / (m K)], and the overall heat transfer coefficient of the substrate is β [W / (m ２ The substrate with a heater according to claim 1 , wherein the following relationship is satisfied: (t / λ)<(1 / β)×0.

10.

8. The substrate with a heater element according to claim 1, wherein the thickness of the substrate is 2 mm or more and 19 mm or less.

9. The substrate with a heater element according to claim 1, wherein the substrate comprises a glass plate.

10. The substrate with a heater according to claim 1, wherein the substrate includes a plurality of layers having different Young's moduli or average linear expansion coefficients, and c1 in the conditional formula is the largest value among the c1 values ​​of the plurality of layers.

11. The substrate with a heater element according to any one of claims 1 to 10, wherein the substrate is a glass plate for a building.

12. The substrate with a heater element according to any one of claims 1 to 10, wherein the substrate is a glass plate for a vehicle.

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