CN113361145B - Method for determining thermal stress of rectangular sheets with different tensile and compression moduli - Google Patents

Method for determining thermal stress of rectangular sheets with different tensile and compression moduli Download PDF

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CN113361145B
CN113361145B CN202110813090.6A CN202110813090A CN113361145B CN 113361145 B CN113361145 B CN 113361145B CN 202110813090 A CN202110813090 A CN 202110813090A CN 113361145 B CN113361145 B CN 113361145B
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compression
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CN113361145A (en
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何晓婷
文思睿
石斌斌
李雪
孙俊贻
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Chongqing University
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Abstract

The invention discloses a method for determining the thermal stress of rectangular sheets with different modulus by tension and compression, which comprises the following steps: modulus of elasticity in tension of E for a block + Modulus of elasticity under compression of E ‑ The surfaces of two ends or one end of the rectangular thin plate with different tensile and compression modulus and with the length of 2l, the height of 2h, the thickness of T and the thermal expansion coefficient of alpha are uniformly heated, so that the temperature rise T (y) of the rectangular thin plate at any section of the rectangular thin plate parallel to the two end surfaces in the height direction is uniformly distributed, wherein y represents the distance from the section of any section of the rectangular thin plate parallel to the two end surfaces in the height direction to the geometric median plane bisecting the height of the rectangular thin plate, and the value interval of y is [ -h, h]Then, based on the stress analysis of the rectangular sheet with different tensile and compressive modulus under the action of the temperature field T (y), the thermal stress σ (y) of the rectangular sheet under the action of the temperature field T (y) can be determined by using the measurement data of the temperature field T (y).

Description

Method for determining thermal stress of rectangular sheets with different tensile and compression moduli
Technical Field
The invention relates to a method for determining thermal stress of rectangular sheets with different modulus under the action of a temperature field.
Background
Sheets are largely used in engineering applications as structures or structural members, while sheets in mechanical engineering, especially in aerospace engineering, are often used in a hot environment. It is well known that most engineering materials have the physical properties of expanding when heated and contracting when cooled. Therefore, if the material composing the thin plate has strong expansion and contraction properties, the temperature change of the thin plate can cause the thin plate to generate obvious expansion and contraction phenomena. If this expansion and contraction is not completely free to expand and contract, so-called thermal stress is usually generated. Thermal stress is also referred to as temperature-changing stress, and external restraint of an object (e.g., a sheet), or mutual restraint between internal portions, is responsible for the thermal stress. It is clear that if the sheet has no external constraints, the thermal stresses inside the sheet will eventually equal zero when the sheet is completely changed from one temperature to another, i.e. the temperature change of the sheet is completely over; on the contrary, if the displacement of the four sides, or the opposite sides, or the upper and lower surfaces of the sheet is externally constrained, the thermal stress inside the sheet will eventually reach a value that is continuously stable and not equal to zero when the temperature variation of the sheet is completely over. It follows that the external restraint of the sheet is the main cause of the sheet developing sustained thermal stresses. The sheets in mechanical engineering or aerospace engineering are often subjected to external constraints such as supports, connections and the like, in other words, the sheets often have continuous and stable thermal stress when the temperature change of the sheets is completely finished. It is clear that if the temperature variation of the sheet is alternating, i.e. the temperature varies periodically, the thermal stress of the sheet is also alternating, i.e. the sheet will be under cyclic thermal stress. In addition to the design problems related to the statics, dynamics and stability of the thin plate, the thermal stress of the thin plate becomes another important problem that must be considered in designing the structure of the thin plate. In short, the structural problems caused by the thermal stress of the thin plate are receiving attention and attention in the design and manufacture of devices, instruments, equipment, and the like in mechanical engineering, especially aerospace engineering.
On the other hand, many engineering materials, such as ceramics, alloys, plexiglass, graphite, etc., typically exhibit significant mechanical properties of different moduli in tension and compression under external forces, i.e., the elastic moduli of the materials in tension and compression are not the same. Most engineering designs, lacking design theory considering the mechanical properties of different tensile and compressive moduli of the material, are generally engineered using the same-modulus design theory, i.e., a design theory obtained under the assumption that the tensile and compressive elastic moduli are equal. Obviously, the design of engineering materials with obviously different modulus mechanical properties by using the same modulus assumption is not reasonable, which brings large calculation errors. However, considering the mechanical properties of the materials with different tensile and compressive moduli increases the nonlinear strength of the mechanical problem under consideration, thereby making it difficult to give a design theory considering different tensile and compressive moduli. From the literature, new results are obtained, and no analytic solution of the sheet thermal stress problem obtained under the condition of considering the mechanical properties of different tensile and compression moduli of the material exists so far. Therefore, it is certainly a very valuable and meaningful task if an analytical solution to the problem of sheet thermal stress can be given under consideration of the mechanical properties of the tensile and compressive modulus of the material.
Disclosure of Invention
The invention is dedicated to the analytic research of the thermal stress problem of the rectangular sheets with different modulus in tension and compression, obtains the analytic solution of the elastic mechanics problem of the rectangular sheets with different modulus in tension and compression under the action of a temperature field, and provides a method for determining the thermal stress of the rectangular sheets with different modulus in tension and compression under the action of the temperature field on the basis.
A method for determining the thermal stress of rectangular sheets with different tensile and compression moduli comprises the following steps: modulus of elasticity in tension of E for a block + Elastic modulus under compression of E - The surfaces of two ends or one end of the rectangular thin plate with different tensile and compression modulus and with the length of 2l, the height of 2h, the thickness of T and the thermal expansion coefficient of alpha are uniformly heated, so that the temperature rise T (y) of the rectangular thin plate at any section of the rectangular thin plate parallel to the two end surfaces in the height direction is uniformly distributed, wherein y represents the distance from the section of any section of the rectangular thin plate parallel to the two end surfaces in the height direction to the geometric median plane bisecting the height of the rectangular thin plate, and the value interval of y is [ -h, h]Then, based on the stress analysis of the rectangular sheet with different tensile and compressive moduli under the action of the temperature field T (y), the analytical relationship between the thermal stress σ (y) of the rectangular sheet at any one of the sections thereof parallel to the two end surfaces in the height direction thereof and the temperature field T (y) can be obtained as
Figure BDA0003168938460000021
Wherein the content of the first and second substances,
Figure BDA0003168938460000022
Figure BDA0003168938460000023
Figure BDA0003168938460000024
Figure BDA0003168938460000025
Figure BDA0003168938460000026
thus, the thermal stress σ (y) of the rectangular thin plate at any one of the cross sections thereof parallel to the both end surfaces in the height direction thereof under the temperature field T (y) can be determined as long as the temperature field T (y) is measured, wherein l, h 1 、h 2 、h、y、y 0 T is in units of millimeters (mm), I + 、I - All units of (A) are cubic millimeters (mm) 4 )，σ、E + 、E - All units of (2) are Newton per square millimeter (N/mm) 2 ) T is given in degrees Celsius (. Degree. C.) and α is given in degrees Celsius (1/C.).
Drawings
FIG. 1 is a schematic diagram showing the elastic mechanics problem of the tension-compression rectangular thin plate with different modulus under the action of the temperature field T (y), wherein 1 represents the tension-compression rectangular thin plate with different modulus, 2 represents the geometric median plane bisecting the heights of the tension-compression rectangular thin plates with different modulus, 3 represents the neutral layer along the height direction of the tension-compression rectangular thin plate with different modulus, 2h represents the height of the tension-compression rectangular thin plate with different modulus, 2l represents the length of the tension-compression rectangular thin plate with different modulus, and T represents the tension-compression rectangular thin plate with different modulusThickness of the plate, h 1 The height h of a tensile region of rectangular sheets with different modulus in tension and compression 2 Indicating the height of the compression zone, y, of rectangular sheets of different moduli under tension 0 Representing the distance from the neutral layer of the rectangular sheet with different modulus along the height direction to the geometric median plane bisecting the height of the rectangular sheet with different modulus, E + Representing the modulus of elasticity in tension, E, of rectangular sheets of different moduli in tension and compression - The compression elastic modulus of the rectangular sheets with different moduli is shown, o is the origin of a coordinate system o-xyz (located at the centroid (i.e. geometric center) of the rectangular sheets with different moduli), x is the coordinate along the length direction of the rectangular sheets with different moduli, y is the coordinate along the height direction of the rectangular sheets with different moduli (used for showing the distance from the section of any one of the rectangular sheets with different moduli parallel to the two end surfaces along the height direction to the geometric middle plane bisecting the heights of the rectangular sheets with different moduli), z is the coordinate along the thickness direction of the rectangular sheets with different moduli, T (y) represents the temperature field acting on the rectangular sheets with different moduli, and the hatched part surrounded by the dotted line indicates the distribution of the temperature field T (y) along the height direction of the rectangular sheets with different moduli.
Detailed Description
The technical scheme of the invention is further explained by combining the specific cases as follows:
as shown in fig. 1, the modulus of elasticity E in tension + ＝45N/mm 2 Modulus of elasticity under compression E - ＝30N/mm 2 Length 2l =10000mm, height 2h =800mm, thickness t =200mm, thermal expansion coefficient α =1.2 × 10 -5 The lower end surface of the rectangular sheet with different tension and compression modulus at/DEG C is uniformly heated along the height direction, so that the temperature rise T (y) of the rectangular sheet at any section of the rectangular sheet parallel to the two end surfaces along the height direction is uniformly distributed, wherein y represents the distance from the section of any one rectangular sheet parallel to the two end surfaces along the height direction to the geometric median plane bisecting the height of the rectangular sheet, and the value range of y is [ -400mm,400mm]Measuring the temperature field T (y) =50-y/8, adopting the method provided by the invention, and obtaining the temperature field T (y) =50-y/8 through the equation
Figure BDA0003168938460000031
Figure BDA0003168938460000032
Figure BDA0003168938460000033
Figure BDA0003168938460000034
Figure BDA0003168938460000041
To obtain h 1 ＝359.59mm、h 2 ＝440.41mm、y 0 ＝40.41mm、I + ＝6.90×10 9 mm 4 、I + ＝1.03×10 10 mm 4 Then using the temperature field T (y) =50-y/8, from the equation
Figure BDA0003168938460000042
σ (y) =0.009-0.00004979y is obtained such that the thermal stress of the rectangular thin plate at a cross section of any one of the two end surfaces thereof parallel to the height direction thereof under the action of the temperature field T (y) =50-y/8 is determined by the equation σ (y) =0.009-0.00004979 y.

Claims (1)

1. A method for determining the thermal stress of rectangular sheets with different modulus in tension and compression is characterized in that: modulus of elasticity in tension of E + Modulus of elasticity under compression of E - The surfaces of two ends or one end of the rectangular thin plate with the length of 2l, the height of 2h, the thickness of t and the thermal expansion coefficient of alpha along the height direction are uniformly heated, so that the rectangular thin plate is uniformly heatedThe temperature rise T (y) of the rectangular thin plate at any section parallel to the two end surfaces in the height direction is uniformly distributed, wherein y represents the distance from any section parallel to the two end surfaces in the height direction to the geometric median plane bisecting the height of the rectangular thin plate, and the value range of y is [ -h, h]Then based on the stress analysis of the rectangular sheet with different modulus under the action of the temperature field T (y), the equation
Figure FDA0003168938450000011
Figure FDA0003168938450000012
Figure FDA0003168938450000013
Figure FDA0003168938450000014
Figure FDA0003168938450000015
Determining h 1 、h 2 、y 0 、I + 、I - Then using the measured data of the temperature field T (y), from the equation
Figure FDA0003168938450000016
Determining the thermal stress sigma (y) of the rectangular thin plate under the action of the temperature field T (y) at any section of the rectangular thin plate parallel to the two end surfaces along the height direction of the rectangular thin plate, wherein l, h 1 、h 2 、h、y、y 0 T is in units of millimeters (mm), I + 、I - All units of (A) are cubic millimeters (mm) 4 )，σ、E + 、E - All units of (2) are Newton per square millimeter (N/mm) 2 ) T is given in degrees Celsius (. Degree. C.) and α is given in degrees Celsius (1/C.).
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