CN114491850B - Method for predicting ultimate curling radius of bean pod rod made of foldable composite material - Google Patents
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- 239000002131 composite material Substances 0.000 title claims abstract description 141
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- 235000010627 Phaseolus vulgaris Nutrition 0.000 title claims abstract description 27
- 244000046052 Phaseolus vulgaris Species 0.000 title claims abstract description 27
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- 230000008859 change Effects 0.000 claims description 3
- 238000005096 rolling process Methods 0.000 claims 2
- 239000000463 material Substances 0.000 abstract description 4
- 238000002788 crimping Methods 0.000 description 5
- 238000004364 calculation method Methods 0.000 description 4
- 238000004026 adhesive bonding Methods 0.000 description 2
- 210000004027 cell Anatomy 0.000 description 2
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- 229920000049 Carbon (fiber) Polymers 0.000 description 1
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- 230000001070 adhesive effect Effects 0.000 description 1
- 239000004917 carbon fiber Substances 0.000 description 1
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- VNWKTOKETHGBQD-UHFFFAOYSA-N methane Chemical compound C VNWKTOKETHGBQD-UHFFFAOYSA-N 0.000 description 1
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Abstract
A method for predicting the ultimate curl radius of a collapsible composite pod rod, the method comprising the four major steps of: step one, defining the geometric shape and size of a foldable composite material bean pod rod, and determining a mathematical expression of the relation among all geometric parameters; secondly, according to a classical laminated board theory and a stress component coordinate transformation equation in the laminated board, the maximum stress in the main direction of the laminated board is obtained; step three, solving the limit curling radius of the foldable composite material bean pod rod in the folding process by using a maximum stress criterion; the method is convenient and efficient, and the limit curling radius of the foldable composite material bean pod rod can be conveniently and quickly predicted only by determining the performance parameters and the geometric parameters of the component materials.
Description
Technical Field
The invention provides a method for predicting the ultimate curling radius of a foldable composite material bean pod rod, and belongs to the field of manned spaceflight.
Background
The foldable composite pod rod has the characteristics of light weight, high rigidity, high folding efficiency, reliable unfolding process and the like, so that the foldable composite pod rod is widely concerned and researched in the field of aerospace, and has a good application prospect. The foldable composite material pod rod is usually made of carbon fiber resin matrix composite materials, and is a thin-wall tubular rod structure capable of realizing folding and unfolding functions. When the folding mechanism is folded, the two ends of the reel are applied with limit curling radiuses to roll up the foldable composite material bean pod rods to form a folded state; when the foldable composite material bean pod rod is unfolded, the foldable composite material bean pod rod can be restored to an unfolded state from a folded state by means of elastic strain energy of the foldable composite material bean pod rod. The structure of the foldable composite material bean pod rod is complicated to deform in the folding process, and the foldable composite material bean pod rod is changed into a flat shape from a bean pod shape when seen from the cross section direction of the foldable composite material bean pod rod; when viewed axially, the collapsible composite pod rod is rolled from the elongated shape to be stowed inside the spacecraft. The superposition of these two deformations is very likely to cause the collapsible composite pod rods to break. Therefore, it is necessary to analyze the limit curl radius. The experiment means directly measures the limit curling radius of the foldable composite material bean pod rod, the cost is high, and the test process is easily influenced by many accidental factors. The numerical simulation method needs to establish a complex finite element model, and has complex calculation, low calculation efficiency and difficult guarantee of calculation precision. Thus, based on classical laminate theory and the maximum stress failure criterion, a method for effectively predicting the ultimate crimp radius of a collapsible composite pod rod is established herein. Only a small amount of component material performance parameters and geometric parameters are needed to quickly and accurately predict the limit curling radius of the foldable composite material bean pod rod, and the method has important academic value and wide engineering application prospect.
Disclosure of Invention
The invention establishes a method for predicting the ultimate curling radius of a pod rod made of a foldable composite material, which has the advantages of simple and convenient calculation, high precision and the like, and the technical scheme is as follows:
the method comprises the following steps of firstly, defining the geometric shape and the size of a foldable composite material pod rod, and determining a mathematical expression of the relation among all geometric parameters.
The collapsible composite pod rods, which are typically comprised of two axisymmetric, curved composite shells, can be collapsed through the cylinder, and are all coiled around the cylinder, taking up little space for storage, as shown in fig. 1. In use, the collapsible composite pod rod may return to its original configuration using elastic strain energy stored by the self-curling deformation. To characterize the geometric properties of a collapsible composite pod stem during the folding deformation, the following basic assumptions were made herein:
(1) Ignoring the variation in wall thickness of the collapsible composite pod rods during the collapsing deformation, the overall deformation can therefore be described by the variation in shape and radius of curvature of the neutral plane, which is not stretched.
(2) The foldable composite pod rod is similar to a thin-wall curved beam, the cross section of the foldable composite pod rod is provided with two symmetrical shafts, and the foldable composite pod rod consists of tangent concave-convex circular arcs with equal central angles.
(3) The folding deformation of the collapsible composite pod rods is idealized as a linear superposition of quasi-static flattening deformation and crimping deformation, equal in both the longitudinal and transverse directions.
With the above assumptions, a polar coordinate system is selected as shown in fig. 2 and 3, and to describe the geometry of the collapsible composite pod rods during crush and crimp deformation, microbodies are selected as shown in fig. 2 and 3. As can be seen in fig. 2, the collapsing deformation of the collapsible composite pod rods may be described by the central angle corresponding to the neutral axis of the cross-section in polar coordinates, and the crimping deformation may be described by the longitudinal curvature of the neutral axis after the entire body in polar coordinates has been fully collapsed. Thus, the following geometric model of the folding deformation of the foldable composite pod rods can be obtained.
According to the assumptions (1) and (2), the constraint condition to be satisfied between the geometric dimensions is
Wherein r is 1 And r 2 Respectively are the curvature radiuses of the tangent concave-convex circular arcs,
is the central angle of the tangent concave-convex circular arc.
The half-piece foldable composite material pod rod consists of two sections of concave circular arcs, one section of convex circular arc and two straight glue-connected edges. Thus, the fully collapsed width b of the collapsible composite pod rod is
Wherein a is the width of the gluing interface.
The cross section area of the foldable composite material bean pod rod is
Wherein t is the wall thickness of the composite shell.
And step two, according to the classical laminated board theory and a stress component coordinate transformation equation in the laminated board, the maximum stress in the main direction in the laminated board is obtained.
The curvature of the concave-convex arc is respectively from 1/r in the process that the foldable composite material pod rod is in a completely flattened state from an initial state 1,0 And 1/r 2,0 Becomes 0 as shown in fig. 2. The arcs with more curvature change during flattening produce more strain and thus more stress. Thus, during collapse of the collapsible composite pod rod, the curvature of the arc of lesser radius of curvature changes to
Wherein,
r＝min{r 1，0 ,r 2，0 } (5)
wherein r is 1,0 And r 2,0 Respectively, the initial curvature radius of the tangent concave-convex circular arc.
When the collapsible composite pod rods are crimped at the extreme crimp radius (as shown in FIG. 3), the curvature in the x-direction changes to
Wherein R is ultimate The extreme crimp radius of the collapsible composite pod rod.
The stress-strain relationship of the kth layer in the laminate is
By substituting and simplifying the formula (4) and the formula (6) into the formula (7)
The principal direction maximum stress of the k-th layer in the laminate can be expressed as the stress component coordinate transformation equation for the k-th layer in the laminate
By substituting the formulae (8) to (10) into the formula (11)
And step three, solving the limit curling radius of the foldable composite material pod rod in the folding process by using a maximum stress criterion.
The critical stress state of the kth layer composite material which has 1-direction tensile failure is
Substituting formula (12) into formula (15) can yield the ultimate crimp radius corresponding to the tensile failure in the direction of the k-th layer composite 1.
The critical stress state of the k-th layer composite material with 1-direction compression failure is
Substituting equation (12) into equation (17) can obtain the ultimate crimp radius corresponding to the compressive failure in the k-th layer composite 1 direction.
The critical stress state of the kth layer composite material which has 2-direction tensile failure is
Substituting formula (13) into formula (19) can obtain the limit curl radius corresponding to the tensile failure in the k-th layer composite material 2 direction.
The critical stress state of the k-th layer composite material with 2-direction compression failure is
Substituting equation (13) into equation (21) can obtain the limit curl radius corresponding to the compressive failure in the k-th layer composite material 2 direction.
The critical stress state of the k-th layer composite material with shear failure is
Substituting equation (14) into equation (23) can yield the ultimate crimp radius corresponding to shear failure.
Wherein, X t And X c Respectively, the longitudinal tensile strength and the compressive strength of the composite material, Y t And Y c Respectively the transverse tensile strength and the compressive strength of the composite material, S 12 Is the composite shear strength.
The maximum value of the ultimate curling radii corresponding to the five failure modes and the n layers of the composite materials is the ultimate curling radius of the pod rod made of the foldable composite material.
The invention relates to a method for predicting the ultimate curling radius of a foldable composite material bean pod rod, which is characterized in that the ultimate curling radius of the foldable composite material bean pod rod can be conveniently and rapidly predicted according to the material performance parameters and the geometric parameters of the components of the foldable composite material bean pod rod.
Drawings
FIG. 1 is a schematic view of a process for folding a collapsible composite pod rod.
FIG. 2 is a schematic representation of the geometry and representative unit cells of a collapsible composite pod rod during collapse.
FIG. 3 is a schematic representation of the geometry and representative unit cells of a collapsible composite pod rod during the crimping process.
The symbols in the figures are as follows:
in FIG. 1: 1. foldable composite pod rods, 2. Cylinders.
In FIG. 2, a is the width of the adhesive interface, P c For compressive loading, r 1 And r 2 Respectively are the curvature radiuses of the tangent concave-convex circular arcs,
is the central angle of the tangent concave-convex circular arc, x, y and z are coordinate axes of a rectangular coordinate system, t is the wall thickness of the composite shell, and delta r 1 Increment of concave arc in wall thickness direction, Δ r 2 The increment of the convex arc along the wall thickness direction.
R in FIG. 3 ultimate Is the limiting crimp radius, Δ R ultimate Is the increment of the double-layer thin shell along the wall thickness direction in the curling process.
The specific implementation mode is as follows:
the method comprises the following steps of firstly, defining the geometric shape and the size of a foldable composite material pod rod, and determining a mathematical expression of the relation among all geometric parameters.
The collapsible composite pod rods, which are typically comprised of two axisymmetric, curved composite shells, can be collapsed through the cylinder, and are all coiled around the cylinder, taking up little space for storage, as shown in fig. 1. In use, the collapsible composite pod rod may be returned to an initial configuration using elastic strain energy stored by self-curling deformation. To characterize the geometric properties of a collapsible composite pod stem during the folding deformation, the following basic assumptions were made herein:
(1) Ignoring the variation in wall thickness of the collapsible composite pod rods during the collapsing deformation, the overall deformation can therefore be described by the variation in shape and radius of curvature of the neutral plane, which is not stretched.
(2) The pod rod made of foldable composite material is similar to a thin-wall curved beam, the cross section of the pod rod is provided with two symmetrical axes, and the pod rod is composed of tangent concave-convex arcs with equal central angles.
(3) The folding deformation of the collapsible composite pod rods is idealized as a linear superposition of quasi-static flattening deformation and crimping deformation, equal in both the longitudinal and transverse directions.
With the above assumptions, a polar coordinate system is selected as shown in fig. 2 and 3, and to describe the geometry of the collapsible composite pod rods during crush and crimp deformation, microbodies are selected as shown in fig. 2 and 3. As can be seen in fig. 2, the collapsing deformation of the collapsible composite pod rods may be described by the central angle corresponding to the neutral axis of the cross-section in polar coordinates, and the crimping deformation may be described by the longitudinal curvature of the neutral axis after the entire body in polar coordinates has been fully collapsed. Thus, the following geometric model of the folding deformation of the foldable composite pod rod can be obtained.
According to the assumptions (1) and (2), the constraint conditions to be satisfied between the geometric dimensions are
Wherein r is 1 And r 2 Respectively are the curvature radiuses of the tangent concave-convex circular arcs,
is the central angle of the tangent concave-convex circular arc.
The half-piece foldable composite material pod rod consists of two sections of concave circular arcs, one section of convex circular arc and two straight glue-connected edges. Thus, the fully collapsed width b of the collapsible composite pod rod is
Wherein a is the width of the gluing interface.
The cross section of the pod rod made of the foldable composite material has the area
Wherein t is the wall thickness of the composite shell.
And step two, according to the classical laminated board theory and a stress component coordinate transformation equation in the laminated board, the maximum stress in the main direction in the laminated board is obtained.
The curvatures of the concave-convex circular arcs are respectively from 1/r in the process of the foldable composite material bean pod rod from the initial state to the completely flattened state 1,0 And 1/r 2,0 Becomes 0 as shown in fig. 2. The arcs with more curvature change during flattening produce more strain and thus more stress. Thus, during collapse of the collapsible composite pod rod, the curvature of the arc of lesser radius of curvature changes to
Wherein,
r＝min{r 1，0 ,r 2，0 } (5)
wherein r is 1,0 And r 2,0 Respectively, the initial curvature radius of the tangent concave-convex circular arc.
When the collapsible composite pod rods are crimped at the limit crimp radius (as shown in FIG. 3), the curvature in the x-direction changes to
Wherein R is ultimate The ultimate radius of curl for the collapsible composite pod rod.
The stress-strain relationship of the kth layer in the laminate is
Substituting the formula (4) and the formula (6) into the formula (7) and simplifying the reaction
The principal direction maximum stress of the k-th layer in the laminate can be expressed as the stress component coordinate transformation equation for the k-th layer in the laminate
By substituting the formulae (8) to (10) into the formula (11)
And step three, solving the limit curling radius of the foldable composite material pod rod in the folding process by using a maximum stress criterion.
The critical stress state of the k-th layer composite material with 1-direction tensile failure is
Substituting formula (12) into formula (15) can yield the ultimate crimp radius corresponding to the tensile failure in the direction of the k-th layer composite 1.
The critical stress state of the k-th layer composite material with 1-direction compression failure is
Substituting equation (12) into equation (17) can obtain the ultimate crimp radius corresponding to the compressive failure in the k-th layer composite 1 direction.
The critical stress state of the k-th layer composite material with 2-direction tensile failure is
Substituting formula (13) into formula (19) can yield the ultimate crimp radius corresponding to the tensile failure in the direction of the k-th layer composite 2.
The critical stress state of the k-th layer composite material which has 2-direction compression failure is
Substituting equation (13) into equation (21) can obtain the limit curl radius corresponding to the compressive failure in the k-th layer composite 2 direction.
The critical stress state of the k-th layer composite material which has shear failure is
Substituting equation (14) into equation (23) can yield the ultimate crimp radius corresponding to shear failure.
Wherein X t And X c Respectively, the longitudinal tensile strength and the compressive strength of the composite material, Y t And Y c Respectively the transverse tensile strength and the compressive strength of the composite material, S 12 Is the composite shear strength.
The maximum value of the ultimate curling radii corresponding to the five failure modes and the n layers of the composite materials is the ultimate curling radius of the pod rod made of the foldable composite material.
The invention relates to a method for predicting the ultimate curling radius of a foldable composite material pod rod, which is characterized in that the ultimate curling radius of the foldable composite material pod rod can be conveniently and quickly predicted according to the performance parameters and the geometric parameters of component materials of the foldable composite material pod rod.
Claims (1)
1. A method of predicting the ultimate curl radius of a collapsible composite pod rod, comprising: the method comprises the following specific steps:
step one, defining the geometric shape and size of a foldable composite material pod rod, and determining a mathematical expression of the relationship among all geometric parameters;
the foldable composite pod rod is generally composed of two axisymmetric and bent composite thin shells, can be folded through a cylinder, is completely coiled on the cylinder, only occupies little space and is convenient to store; when in use, the foldable composite material bean pod rod can be restored to the initial configuration by utilizing the elastic strain energy stored by self-curling deformation; to characterize the geometric properties of a collapsible composite pod stem during the folding deformation, the following basic assumptions were made herein:
(1) Ignoring wall thickness variations of the collapsible composite pod rods during the collapsing deformation, therefore, overall deformation can be described by variations in the shape and radius of curvature of the neutral plane, which is not stretched;
(2) The foldable composite material pod rod is similar to a thin-wall curved beam, the cross section of the foldable composite material pod rod is provided with two symmetrical shafts, and the foldable composite material pod rod consists of tangent concave-convex circular arcs with equal central angles;
(3) The folding deformation of the foldable composite material pod rod is idealized as the linear superposition of quasi-static flattening deformation and curling deformation, and is equal along the longitudinal direction and the transverse direction;
the flattening deformation of the collapsible composite pod rod can be described by a central angle corresponding to a neutral axis of a cross section under a polar coordinate, and the curling deformation can be described by longitudinal curvature of the neutral axis after the whole body under the polar coordinate is completely flattened; therefore, the following geometric model of the folding deformation of the folding composite material bean pod rod can be obtained;
according to the assumptions (1) and (2), the constraint condition to be satisfied between the geometric dimensions is
Wherein r is 1 And r 2 Are respectively the curvature radius of the tangent concave-convex circular arc,
the central angle is a tangent concave-convex arc;
the half-piece foldable composite material bean pod rod consists of two sections of concave circular arcs, one section of convex circular arc and two straight glue-connected edges; thus, the fully collapsed width b of the collapsible composite pod rod is
Wherein a is the width of the cementing interface;
the cross section area of the foldable composite material bean pod rod is
Wherein t is the wall thickness of the composite shell;
secondly, according to a classical laminated board theory and a stress component coordinate transformation equation in the laminated board, the maximum stress in the main direction of the laminated board is obtained;
the curvatures of the concave-convex circular arcs are respectively from 1/r in the process of the foldable composite material bean pod rod from the initial state to the completely flattened state 1,0 And 1/r 2,0 Becomes 0; during the flattening process, the arc with larger curvature change generates larger strain, and then larger stress is generated; thus, during collapse of the collapsible composite pod rod, the curvature of the arc of lesser radius of curvature changes to
Wherein,
r＝min{r 1，0 ,r 2，0 } (5)
wherein r is 1,0 And r 2,0 Initial curvature radii of the tangent concave-convex circular arcs are respectively;
when the collapsible composite pod rod is crimped at the limit crimp radius, the curvature in the x-direction changes to
Wherein R is ultimate The ultimate radius of curl for a collapsible composite pod rod;
the stress-strain relationship of the kth layer in the laminate is
By substituting and simplifying the formula (4) and the formula (6) into the formula (7)
The principal direction maximum stress of the k-th layer in the laminate can be expressed as the stress component coordinate transformation equation for the k-th layer in the laminate
By substituting the formulae (8) to (10) into the formula (11)
Step three, solving the limit curling radius of the foldable composite material bean pod rod in the folding process by using a maximum stress criterion;
the critical stress state of the kth layer composite material which has 1-direction tensile failure is
Substituting the formula (12) into the formula (15) to obtain the limit crimp radius corresponding to the tensile failure of the k-th layer composite material 1 in the direction;
the critical stress state of the k-th layer composite material with 1-direction compression failure is
Substituting the formula (12) into the formula (17) can obtain the limit crimp radius corresponding to the compressive failure of the k-th layer composite material 1 in the direction;
the critical stress state of the k-th layer composite material with 2-direction tensile failure is
Substituting the formula (13) into the formula (19) to obtain the limit crimp radius corresponding to the tensile failure of the k-th layer composite material 2 in the direction;
the critical stress state of the k-th layer composite material with 2-direction compression failure is
Substituting the formula (13) into the formula (21) can obtain the limit crimp radius corresponding to the compressive failure of the k-th layer composite material 2 in the direction;
the critical stress state of the k-th layer composite material with shear failure is
Substituting the formula (14) into the formula (23) can obtain the limit crimp radius corresponding to the shear failure;
wherein, X t And X c Respectively the longitudinal tensile strength and the compressive strength of the composite, Y t And Y c Respectively the transverse tensile strength and the compressive strength of the composite material, S 12 Is the composite shear strength;
the maximum value of the limit rolling radius respectively corresponding to the five failure modes and the n layers of composite materials is the limit rolling radius of the pod rod made of the foldable composite material;
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