CN107764453A - Milling Process piece surface residual stress measuring method based on strain variation and anti-pushing manipulation - Google Patents

Milling Process piece surface residual stress measuring method based on strain variation and anti-pushing manipulation Download PDF

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CN107764453A
CN107764453A CN201710970319.0A CN201710970319A CN107764453A CN 107764453 A CN107764453 A CN 107764453A CN 201710970319 A CN201710970319 A CN 201710970319A CN 107764453 A CN107764453 A CN 107764453A
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mrow
msub
msubsup
centerdot
msup
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孟龙晖
方成刚
王�华
黄筱调
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NANJING GONGDA CNC TECHNOLOGY Co Ltd
Nanjing Tech University
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NANJING GONGDA CNC TECHNOLOGY Co Ltd
Nanjing Tech University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01L—MEASURING FORCE, STRESS, TORQUE, WORK, MECHANICAL POWER, MECHANICAL EFFICIENCY, OR FLUID PRESSURE
    • G01L5/00—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes
    • G01L5/0047—Apparatus for, or methods of, measuring force, work, mechanical power, or torque, specially adapted for specific purposes measuring forces due to residual stresses

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  • General Physics & Mathematics (AREA)
  • Other Investigation Or Analysis Of Materials By Electrical Means (AREA)
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Abstract

The present invention discloses a kind of Milling Process piece surface residual stress measuring method based on strain variation and anti-pushing manipulation, and it belongs to mechanical engineering field.During traditional x-ray method measurement Milling Process piece surface residual stress, what it was measured is all often that residual stress has reached the stress value after poised state, and its equipment is expensive, the present invention is by measuring the change to the strain at its machined surface back side during surface residual stress layer delamination, simultaneously in view of residual stress in residual stress layer after each delamination because the conversion of moment of flexure caused by the change of neutral line position, counted since the bottom, it is progressively up counter to push away, untill the residual-stress value in the superiors calculates, complete the measurement of the surface residual stress caused by whole Milling Process with change in depth.Its required equipment of whole process is cheap, and it is simple to operate, and its value for measuring can more Accurate Prediction be variously-shaped and the stress of size is because deflection and internal stress caused by surface residual stress are finally reached the distribution situation of the actual residual stress of inside parts after balance.

Description

Milling Process piece surface residual stress measurement based on strain variation and anti-pushing manipulation Method
Technical field：
The present invention relates to the Milling Process piece surface of a kind of measurement based on piece surface strain variation and anti-pushing manipulation is residual Residue stress measuring method, its method for measuring stress belonged in mechanical engineering field.
Background technology：
Residual stress, namely internal stress, its property and size can have a strong impact on military service performance and the life-span of part, in industry In, manufacturer often compares the advantageous effect for focusing on improving residual stress, and reduces its illeffects, so as to further improve The quality of product.
In the cutting process to metal parts, final piece surface can be caused to form one layer of residual stress layer, In the stressor layers, the distribution of residual stress is very shallow, typically not over 0.2mm, but but has in the depth direction very high Rate of change, the size and property of its stress value are a key factors of piece surface crudy, and it can influence part Precision (particularly rigidity weaker part), static strength, the generation of fatigue life, corrosion resistance and crackle, thus it is right The research of the stress of the surface residual of machining part is just particularly important.
Research for surface residual stress, often since it is with the measurement of the value of change in depth, at present more into Ripe e measurement technology is x-ray method combination layer stripping, but is distributed in very great Cheng due to the final surface residual stress value of part The rigid influence of part is received on degree, therefore even if identical part material, identical processing method, machined parameters, processes bar Part, because the rigidity of part is inconsistent, it is possible to ultimately result in the stress value after reaching self-balancing and inconsistent phenomenon is presented, because The surface residual stress value for a certain part that this x-ray method combination layer stripping finally measures is not representative.
The formation of the final surface residual stress of part to be processed can logically be distributed two steps：1, cutting process Various factors causes piece surface that plastic deformation and phase transformation occurs, and forms surface residual stress layer；2, it is remaining by piece surface Certain deformation can occur for the influence of stress, part, finally cause surface residual stress redistribution, while whole part is interior Stress reaches balance.It is obvious that the stress value measured with x-ray method is part reached balance after stress value have occurred and that change Surface residual stress after change, preceding value of having addressed is influenceed by detail rigidity, therefore it does not possess representativeness, then how Value of the surface residual stress before self-balancing measured in part is just particularly important, and it eliminates the rigid shadow of part Ring, and the influence of various machined parameters and processing conditions to piece surface residual stress that can fully withdraw deposit.
The content of the invention：
Problem to be solved by this invention is：The defects of for prior art, propose that a kind of strain variation that is based on pushes away with counter The Milling Process piece surface residual stress measuring method of method, realizes the measurement of value of the residual stress before self-balancing is reached, Eliminate traditional x-ray method measured value in response to force self-balanced influence caused error, its final obtained value can be pre- The part of survey any shape and size deflection caused by because of surface residual stress and the stress value after final self-balancing.
The present invention adopts the following technical scheme that：Milling Process piece surface residual stress based on strain variation and anti-pushing manipulation Measuring method, it comprises the following steps：
(1) part is first made annealing treatment to remove the internal stress of its own, in case the internal stress of its own can be to rear The measurement result of phase produces certain influence；
(2) Milling Process processing is carried out to the part after annealing；
(3) two-way foil gauge is sticked at the back side of the machined surface of part, is respectively used for measuring X and Y-direction, be i.e. milling cutter Axial length direction and the strain of direction of feed；
(2) successively delamination is carried out to machined surface, after often shelling layer of material, records X and Y-direction that foil gauge measures respectively Strain change, and measure and record the thickness for the material layer that each delamination removes, the foil gauge after continuing delamination simultaneously The strain value measured continues constant；
(3) it is n-th this delamination that setting strains the delamination changed for the last time, then thinks foil gauge after n-th delamination The strain that measures entirely due to the release of residual stress in n-th layer and cause, X and Y-direction in n-th layer can be calculated to obtain Residual-stress value be respectivelyWith
(4) after the X and the stress of Y-direction for obtaining n-th layer, it is based onWithFoil gauge measures after (n-1)th delamination The change of strainAnd the thickness h of delamination twicen, hn-1, the residual thickness H meters of part also after n-th delamination Calculate (n-1)th layer of residual stress be respectively
(5) calculate n-th layer residual stress when, based on it is counted from N+1 layer to n-layer in residual-stress value, And the thickness of corresponding delamination every time, the change of the strain measured after each delamination, it may finally calculate residual in n-th layer Residue stress value；
(6) above step is repeated, until the residual-stress value in the 1st layer is tried to achieve, now the remnants in all stressor layers should Force value is tried to achieve completely, i.e., has been obtained completely with the Milling Process piece surface residual stress of change in depth.
The residual stress in n-th layer and (n-1)th layer is calculated with below equation:
X and the stress of Y-direction in n-th layerRespectively：
Wherein E'=E/ (1- μ2), E and μ are respectively the modulus of elasticity and Poisson's ratio of part material,WithRespectively The milling cutter direction of feed and the change of the strain of axial direction that foil gauge measures after n-th delamination, hnFor the thickness of n-th delamination, H For the residual thickness of part after n-th delamination；
X and the stress of Y-direction in (n-1)thRespectively:
WhereinWithThe change for the strain that foil gauge measures, h after respectively (n-1)th delaminationn-1For (n-1)th time The thickness of delamination.
The residual stress in N (1≤N≤n-2) layer is calculated with below equation：
Wherein, in formula (75)
Pn-N+1=hn-N+1·(4·H-2·hn-2·hn-1-……-2·hn-N+2+hn-N+1)
……
……
In formula (76)
……
……
In above formula,Represent the change of the strain of foil gauge measures after the A times delamination X and Y-direction, hA Subscript represent the removal of the A time delamination material layer thickness,Represent in the layer for the stressor layers that the A times delamination removes X and Y-direction residual stress.
The present invention has the advantages that：
(1) thickness of each delamination is measured by the change of foil gauge measuring strain and feeler, measurement required for it is set It is standby that relative to X ray stress gauge, its value is very cheap,；
(2) its stress measured is that residual stress reaches value before self-balancing state, therefore it eliminates the shape of part The influence of shape and size to self-balancing result, and the stress value before the self-balancing state can predict that part is residual because of the surface The distribution situation of its internal stress when reaching self-balancing state after deformation caused by residue stress and part deformation；
(3) it is simple to operate, without special training and special technical ability.
Brief description of the drawings：
Fig. 1 is to stick two-way foil gauge schematic diagram at the back side of the machined surface of part.
Fig. 2 is that part is stripped only remaining last layer i.e. schematic diagram of n-th layer.
Fig. 3 is that part is stripped only remaining i.e. last two layers n-th layer and (n-1)th layer of schematic diagram.
Wherein：
1-Y is to strain measurement foil gauge, and 2-X is to strain measurement foil gauge, 3- binding posts.
Embodiment：
Technical scheme is described in detail below in conjunction with accompanying drawing.
Two-way foil gauge is sticked at the back side of the machined surface of part first, is corroding stripping to residual stress layer for measuring The change of the strain of the X at its back side and Y-direction during layer.
The calculating of moment of flexure is based primarily upon the size and the stressor layers of stressor layers internal stress caused by residual stress It is calculated with the distance of neutral line, neutral line is always positioned at the position among the geometry of part, is each erosion removal After layer of material, the position of the neutral line of part can all occur to change accordingly, therefore often after removal layer of material, remainder Stress changed due to the distance of itself and neutral line, therefore its caused moment of flexure can also occur to change accordingly, therefore When stressor layers do not remove completely to be finished, per erosion removal layer of material, the change of the moment of flexure of part is not entirely due to gone Cause except the stress in layer, but the result ultimately formed with the comprehensive function of the stress of remainder.
It is set in after eliminating n-layer altogether and measures part there is no the change of strain, now can be concluded that and removing the n-th material Residual stress layer is all removed and finished after the bed of material.After n-th layer material is being removed, the change of the internal stress and moment of flexure of part It is considered that entirely due to stress σ in n-th layernRelease caused by, can be here n-th from last layer therefore The X of layer and the stress of Y-direction start to count.
The depth of the material layer removed when n-th layer delamination is set as hn, the residual thickness of part is H, such as Fig. 1 after removal It is shown.It is consistent that every layer of internal stress is set in calculating process.When it is n-th layer to be only left last layer, neutral line Stress is respectively:
WhereinWithThe residual stress of X and Y-direction respectively in n-th layer, the strain that neutral line removes are respectively：
Wherein E and μ is respectively the modulus of elasticity and Poisson's ratio of part material, and the curvature for setting X and Y-direction is respectively：
The strain of X and Y-direction can be expressed as：
After equilibration had been achieved,
In the range of, the stress of X and Y-direction is respectively：
Wherein E'=E/1- μ2,In the range of, the stress of X and Y-direction is respectively：
According to bending moment self-balancing principle, the moment of flexure of X and Y-direction is respectively：
It can further be expressed as：
It can obtain：
When it is n-th layer to remove last layer, the change of the X for the upper surface that foil gauge measures and the strain of Y-directionWithIt can be expressed as respectively：
Can be by curvatureWithIt is expressed as with the strain measured：
It can be obtained according to formula (15) and (16)：
Finally try to achieve X and the stress of Y-direction in n-th layerRespectively：
So far, the stress value of last layer has been obtained, and seeks the layer second from the bottom i.e. stress value of n-1 layers below.
(n-1)th layer of the thickness removed is hn-1, as shown in Figure 3.Only it is being left last two layers, i.e. n-th layer and (n-1)th layer When, the stress and strain of neutral line is respectively:
WhereinWithThe residual stress of X and Y-direction in respectively (n-1)th layer,
Setting X and the curvature of Y-direction are respectively：
The strain of X and Y-direction can be expressed as：
When reaching balance,
In the range of, the stress of X and Y-direction is respectively：
In the range of, the stress of X and Y-direction is respectively：
In the range of, the stress of X and Y-direction is respectively：
According to bending moment self-balancing principle, the moment of flexure of X and Y-direction is respectively：
It can further be expressed as：
It can then obtain：
When it is (n-1)th layer to remove layer second from the bottom, the change of the X for the upper surface that foil gauge measures and the strain of Y-directionWithIt can be expressed as respectively：
Can be by curvatureWithIt is expressed as with the strain measured：
It can be obtained according to formula (41), (42):
X and the stress of Y-direction in layer second from the bottom i.e. (n-1)th finally askedRespectively:
By that analogy, the calculating process of each layer of stress is all based on answering in those stressor layers of remaining non-delamination Power is calculated.When being only left last N layers, the stress and strain of neutral line is respectively:
Setting X and the curvature of Y-direction are respectively：
The strain of X and Y-direction can be expressed as：
When reaching balance,In the range of, X Stress with Y-direction is respectively：
In the range of, the stress point of X and Y-direction It is not：
……
In the range of, the stress point of X and Y-direction It is not：
In the range of, the stress point of X and Y-direction It is not：
According to bending moment self-balancing principle, the moment of flexure of X and Y-direction is respectively：
It can further be expressed as：
It can then obtain：
When being n-N+1 layers when removing n-th layer reciprocal, foil gauge measures the change of the X of upper surface and the strain of Y-directionWithIt can be expressed as respectively：
Can be by curvatureWithIt is expressed as with the strain measured：
According to formula (69), (70) can obtain：
Wherein, in formula (75)
Pn-N+1=hn-N+1·(4·H-2·hn-2·hn-1-……-2·hn-N+2+hn-N+1)
……
……
In formula (76)
……
……
So far, the residual stress of the X in any one layer and Y-direction is obtained completely, and what it was calculated is all zero Part deforms and make it that its internal stress reaches the stress value before balance, and the rigid influence of part is eliminated from bat.
Described above is only the preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art For member, some improvement can also be made under the premise without departing from the principles of the invention, and these improvement also should be regarded as the present invention's Protection domain.

Claims (5)

1. the Milling Process piece surface residual stress measuring method based on strain variation and anti-pushing manipulation, it is characterised in that：It is wrapped Include following steps：
(1) part is first made annealing treatment to remove the internal stress of its own, in case the internal stress of its own can be to the later stage Measurement result produces certain influence；
(2) Milling Process processing is carried out to the part after annealing；
(3) two-way foil gauge is sticked at the back side of the machined surface of part, is respectively used for measuring X and Y-direction, i.e. milling cutter axial length Direction and the strain of direction of feed；
(2) successively delamination is carried out to machined surface, after often shelling layer of material, record X that foil gauge measures and Y-direction respectively should The change of change, and the thickness for the material layer that each delamination removes is measured and recorded simultaneously, foil gauge measures after continuing delamination Strain value continue it is constant；
(3) it is n-th this delamination that setting strains the delamination changed for the last time, then foil gauge measures after thinking n-th delamination Strain entirely due to the release of residual stress in n-th layer and cause, can calculate X in n-th layer and Y-direction it is residual Residue stress value is respectivelyWith
(4) after the X and the stress of Y-direction for obtaining n-th layer, it is based onWithThe strain that foil gauge measures after (n-1)th delamination ChangeAnd the thickness h of delamination twicen, hn-1, the residual thickness H of part is calculated also after n-th delamination (n-1)th layer of residual stress is respectively
(5) calculate n-th layer residual stress when, based on it is counted from N+1 layer to n-layer in residual-stress value, and The thickness of corresponding delamination every time, the change of the strain measured after each delamination, may finally calculate remnants in n-th layer should Force value；
(6) above step is repeated, until the residual-stress value in the 1st layer is tried to achieve, the now residual-stress value in all stressor layers Try to achieve, i.e., obtained completely with the Milling Process piece surface residual stress of change in depth completely.
2. the Milling Process piece surface residual stress measurement side based on strain variation and anti-pushing manipulation as claimed in claim 1 Method, it is characterised in that the back side of place part machined surface stick two-way foil gauge with and meanwhile measure milling cutter axial direction and direction of feed Strain change.
3. the Milling Process piece surface residual stress measurement side based on strain variation and anti-pushing manipulation as claimed in claim 1 Method, it is characterised in that the residual stress in each layer is calculated with anti-pushing manipulation, i.e., is calculated since the n-th layer of last time delamination, Then the residual stress in (n-1)th layer is calculated, until the residual stress in the 1st layer is obtained.
4. the Milling Process piece surface residual stress measurement side based on strain variation and anti-pushing manipulation as claimed in claim 1 Method, it is characterised in that calculate the residual stress in n-th layer and (n-1)th layer with below equation:
X and the stress of Y-direction in n-th layerRespectively：
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Wherein E'=E/ (1- μ2), E and μ are respectively the modulus of elasticity and Poisson's ratio of part material,WithRespectively n-th The milling cutter direction of feed and the change of the strain of axial direction that foil gauge measures after secondary delamination, hnFor the thickness of n-th delamination, H n-th The residual thickness of part after secondary delamination；
X and the stress of Y-direction in (n-1)thRespectively:
<mrow> <msubsup> <mi>&amp;sigma;</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <msup> <mi>E</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;CenterDot;</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;mu;</mi> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>(</mo> <mi>H</mi> <mo>+</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>x</mi> <mi>n</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mi>H</mi> <mo>+</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>49</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msubsup> <mi>&amp;sigma;</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mfrac> <mrow> <mo>-</mo> <msup> <mi>E</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;CenterDot;</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;mu;</mi> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>(</mo> <mi>H</mi> <mo>+</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>y</mi> <mi>n</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mi>H</mi> <mo>+</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>50</mn> <mo>)</mo> </mrow> </mrow>
WhereinWithThe change for the strain that foil gauge measures, h after respectively (n-1)th delaminationn-1For (n-1)th delamination Thickness.
5. the Milling Process piece surface residual stress measurement side based on strain variation and anti-pushing manipulation as claimed in claim 1 Method, it is characterised in that calculate the residual stress in N (1≤N≤n-2) layer with below equation：
<mrow> <msubsup> <mi>&amp;sigma;</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>M</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>J</mi> <mi>x</mi> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>J</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>J</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msubsup> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>J</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <msup> <mi>P</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>75</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msubsup> <mi>&amp;sigma;</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>M</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>J</mi> <mi>y</mi> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>J</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>J</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msubsup> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>J</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> </mrow> <msup> <mi>P</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mfrac> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>76</mn> <mo>)</mo> </mrow> </mrow>
Wherein, in formula (75)
<mrow> <msubsup> <mi>M</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msup> <mi>E</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;CenterDot;</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;mu;</mi> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>(</mo> <mi>H</mi> <mo>+</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow>
Pn-N+1=hn-N+1·(4·H-2·hn-2·hn-1-……-2·hn-N+2+hn-N+1)
<mrow> <msubsup> <mi>J</mi> <mi>x</mi> <mi>n</mi> </msubsup> <mo>=</mo> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>x</mi> <mi>n</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>+</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
<mrow> <msubsup> <mi>J</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
……
<mrow> <msubsup> <mi>J</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>-</mo> <mo>...</mo> <mo>...</mo> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
……
<mrow> <msubsup> <mi>J</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>x</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>-</mo> <mo>...</mo> <mo>...</mo> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
In formula (76)
<mrow> <msubsup> <mi>M</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msup> <mi>E</mi> <mo>&amp;prime;</mo> </msup> <mo>&amp;CenterDot;</mo> <mo>&amp;lsqb;</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>y</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <mi>&amp;mu;</mi> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mi>n</mi> </msubsup> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msubsup> <mi>&amp;Delta;&amp;epsiv;</mi> <mrow> <mi>m</mi> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>&amp;CenterDot;</mo> <msup> <mrow> <mo>(</mo> <mi>H</mi> <mo>+</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow>
<mrow> <msubsup> <mi>J</mi> <mi>y</mi> <mi>n</mi> </msubsup> <mo>=</mo> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>y</mi> <mi>n</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>+</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
<mrow> <msubsup> <mi>J</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
……
<mrow> <msubsup> <mi>J</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>-</mo> <mo>...</mo> <mo>...</mo> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>...</mo> <mo>...</mo> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
……
<mrow> <msubsup> <mi>J</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo>-</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>y</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msubsup> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <mi>H</mi> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mi>n</mi> </msub> <mo>-</mo> <mo>...</mo> <mo>...</mo> <mo>-</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <mo>&amp;CenterDot;</mo> <msub> <mi>h</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>N</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow>
In above formula,Represent the change of the strain of foil gauge measures after the A times delamination X and Y-direction, hAUnder Mark represents the thickness for the material layer that the A times delamination removes,Represent X in the layer for the stressor layers that the A times delamination removes and The residual stress of Y-direction.
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