CN105022919A - Method for predicting size of low-speed impact dent of circular metal sheet - Google Patents

Abstract

The invention provides a method for predicting the size of a low-speed impact dent of a circular metal sheet. The method has the advantages of simplicity and convenience for calculation, high precision and the like. The technical scheme adopted by the invention is as follows: providing assumed conditions of a new method for predicating the size of the low-speed impact dent of the circular metal sheet; determining a deformation function of the impact dent according to a loaded form and dent features during low-speed impact of the circular metal sheet; further establishing a corresponding stress-strain relation by the provided deformation function of the impact dent of the circular metal sheet; according to a functional principle, establishing a control equation of the circular metal sheet in a low-speed impact condition, solving undetermined parameters of the control equation with a numerical solution method, and at last the size of the impact dent of the metal sheet is very conveniently determined.

Description

A kind of method predicting circular metal thin plate low velocity impact dimple size

Technical field

The invention provides a kind of method predicting circular metal thin plate low velocity impact dimple size, belong to mechanics design field.

Background technology

Sheet metal is often easily subject to sandstone, the unexpected low velocity impact event such as maintenance tool, cause expendable plastic yield, form impact dent, directly affect its static strength and the performance such as tired, therefore, needing to carry out assessing sheet metal by the impact dent size under low velocity impact to this kind of low velocity impact collision phenomenon is one of very crucial input parameter for designer, especially to material selection and size design most important, therefore, how to obtain the impact dent size of sheet metal under low velocity impact is all the problem that engineering staff needs to solve all the time.Its impact dent size directly can be measured by the test of sheet metal low velocity impact, but experimentation cost is higher, and the cycle of wasting time and energy is long, especially in the initial design stage, the impact dent size in any materials, size and impact velocity situation is all determined it is unpractical by test; Numerical simulation method needs to set up complicated finite element model, calculation of complex, and counting yield is low; More existing analytic method more complicated, theory solves also inconvenient.When assessment sheet metal is subject to the impact dent size under low velocity impact, the normal circular metal thin plate of classics that adopts is as the target plate be hit.Therefore, the present invention proposes a kind of analytic method predicting circular metal thin plate low velocity impact dimple size, the method is very simple and practical, only need a small amount of metallic sheet stock parameter and geometric parameter, just can be easy to obtain and the impact dent size under different impact velocity, visible the present invention has Important Academic meaning and engineer applied is worth.

Summary of the invention

The invention provides a kind of new method predicting circular metal thin plate low velocity impact dimple size, it is easy that the method has calculating, precision advantages of higher, and its technical scheme is as follows:

The assumed condition of the new method of step one, proposition prediction circular metal thin plate low velocity impact dimple size.

Assumed condition comprises:

(1) the impact dent shape of sheet metal is rotational symmetric, and does not consider the impact of resilience on dimple size;

(2) meet kirchhoff-Le Fu thin plate hypothesis, therefore can ignore the impact of normal stress and transverse shear stresses outside face, be mainly radial drawing stress, radially bend stress and circumferential skewing stress;

(3) material of sheet metal is elastoplasticity linear strain-hardening material, and as shown in Figure 3, elastic stage, line and be the plastic stage, the slope of its correspondence is respectively E and E to its strain-stress relation ^*, ε _efor elastic limit strain;

(4) ignore air resistance, impact process friction force homenergic dissipates, think that impact energy is all converted into strain energy.

Step 2, according to loading during circular metal thin plate low velocity impact and pit feature, determine the warping function of impact dent.

According to basic assumption (1) above, because impact dent shape has rotational symmetry, therefore, can describe impact dent shape under cylindrical-coordinate system r θ z, wherein, r is radial distance coordinate, θ azimuthal coordinate.In impact dent region, the z in face represents to distortion with w, and known w is only the function about r, and has nothing to do with θ.Circular metal sheet edges is clamped, and central role has centre-point load p, and Ze Ju center is that the concentric circles place of r has total shearing need balance with centre-point load p, specifically can be expressed as

2πrQ _r＝p (1)

In formula, Q _rfor apart from plate center being the shearing at the concentric circles place of r.

Polar coordinates formula according to shearing:

$Q_r=-D\frac{d}{dr}(\frac{d^{2}w}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{dw}{dr})---\left(2\right)$

Formula (2) is substituted in formula (1), can obtain

$-2\mathrm{\π}rD\frac{d}{dr}(\frac{d^{2}w}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{dw}{dr})=p---\left(3\right)$

$-D\frac{d}{dr}(\frac{d^{2}w}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{dw}{dr})=\frac{p}{2\mathrm{\π}r}---\left(4\right)$

To formula (4) integration three times, can obtain

$w=\frac{p}{8\mathrm{\π}D}(r^2\ln r+C_1{r}^2+C_2\ln r+C_3)---\left(5\right)$

In formula, C ₁, C ₂and C ₃for undetermined constant; D is bending stiffness, specifically can be expressed as

$D=\frac{E^{*}{t}_0^3}{12(1-v^2)}---\left(6\right)$

In formula, E ^*for surrender section modulus, v is Poisson ratio, t ₀for the thickness of plate.

Circular metal thin plate clamped constraint time boundary condition need meet:

When r=0 warping function need meet as downstream condition:

${\frac{dw}{dr}|}_{r=0}=f^{\′}\left(0\right)=0---\left(7\right)$

Work as r=r ₀warping function need meet as downstream condition:

$\left\{\begin{array}{c}{w|}_{r=r_0}=f\left(r_0\right)=0\\ {\frac{dw}{dr}|}_{r=r_0}=f^{\′}\left(r_0\right)=0\end{array}\right.---\left(8\right)$

Formula (5) is substituted into respectively in formula (7) and formula (8), can obtain

$\left\{\begin{array}{c}{C}_1=-\frac{1}{2}-{\mathrm{lnr}}_0\\ C_2=0\\ C_3=\frac{1}{2}{r}_0^2\end{array}\right.---\left(9\right)$

Formula (9) is substituted in formula (5), can obtain

$w=\frac{p}{8\mathrm{\π}D}(r^2\ln \frac{r}{r_0}+\frac{r_0^2-r^2}{2})---\left(10\right)$

Step 3, the circular metal thin plate impact dent warping function utilizing step 2 to propose, set up corresponding strain stress relation further.

According to basic assumption (2) above, circular metal thin plate by the deformation process of low velocity impact, mainly by radial-draw deformation, radially bend distortion and circumferential skewing distortion apparatus with shock absorbing.

Circular metal thin plate radial drawing strain stress _rtcan be expressed as

${\mathrm{\ϵ}}_{rt}=\frac{\sqrt{{\mathrm{dw}}^2+{\mathrm{dr}}^2}-dr}{dr}=\sqrt{{\left(\frac{dw}{dr}\right)}^2+1}-1=\frac{1}{2}{\left(\frac{dw}{dr}\right)}^2---\left(11\right)$

Formula (10) is substituted in formula (11), can obtain

${\mathrm{\ϵ}}_{rt}=\frac{p^2}{32{\mathrm{\π}}^2{D}^2}{r}^2{\left(ln\frac{r}{r_0}\right)}^2---\left(12\right)$

Carry out variable replacement, order

$x=\frac{r}{r_0}---\left(13\right)$

The variable of formula (13) is replaced and is equivalent to carry out normalized to variable r, therefore 0≤x≤1.

Formula (13) is substituted in formula (12), can obtain

${\mathrm{\ϵ}}_{rt}=\frac{p^2{r}_0^2}{32{\mathrm{\π}}^2{D}^2}{x}^2{\left(\ln x\right)}^2={\mathrm{Hp}}^2{x}^2{\left(\ln x\right)}^2---\left(14\right)$

In formula, H is constant, can be expressed as

$H=\frac{r_0^2}{32{\mathrm{\π}}^2{D}^2}---\left(15\right)$

Thin plate radially bend curvature κ _rwith circumferential skewing curvature κ _θcan be expressed as

$\left\{\begin{array}{c}{\mathrm{\κ}}_r=\frac{d^{2}w}{{\mathrm{dr}}^2}=\frac{p}{4\mathrm{\π}D}(ln\frac{r}{r_0}+1)\\ {\mathrm{\κ}}_{\mathrm{\θ}}=\frac{1}{r}\left(\frac{dw}{dr}\right)=\frac{p}{4\mathrm{\π}D}\ln \frac{r}{r_0}\end{array}\right.---\left(16\right)$

Formula (13) is substituted in formula (16), can obtain

$\left\{\begin{array}{c}{\mathrm{\κ}}_r=\frac{p}{4\mathrm{\π}D}(\ln x+1)\\ {\mathrm{\κ}}_{\mathrm{\θ}}=\frac{p}{4\mathrm{\π}D}\ln x\end{array}\right.---\left(17\right)$

Thin plate radially bend strain stress _rbwith circumferential skewing strain stress _{θ b}can be expressed as

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{rb}={\mathrm{\κ}}_{r}z=\frac{p}{4\mathrm{\π}D}z(\ln x+1)\\ {\mathrm{\ϵ}}_{\mathrm{\θ}b}={\mathrm{\κ}}_{\mathrm{\θ}}z=\frac{p}{4\mathrm{\π}D}z\ln x\end{array}\right.---\left(18\right)$

Step 4, according to the principle of work and power, set up circular metal thin plate governing equation under low velocity impact condition, recycling method of value solving solves the undetermined parameter of governing equation, finally determines impact dent size.

According to basic assumption (3) above, the strain energy of circular metal thin plate radial-draw deformation can be expressed as

$U_t=\underset{V}{\∫}(\∫{\mathrm{\σ}}_{rt}d\mathrm{\ϵ})dV=\underset{V}{\∫}\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e)+\frac{E^{*}{({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e)}^2}{2}\]dV---\left(19\right)$

Formula (14) is substituted in formula (19), can obtain

$\begin{array}{c}{U}_t={\∫}_0^{r_0}{\∫}_{-\frac{t_0}{2}}^{\frac{t_0}{2}}{\∫}_0^{2\mathrm{\π}}\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s\left({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e\right)+\frac{E^{*}{\left({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e\right)}^2}{2}\]rd\mathrm{\θ}dzdr\\ =2{\mathrm{\πt}}_0{r}_0^2{\∫}_0^1\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s\left({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e\right)+\frac{E^{*}{\left({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e\right)}^2}{2}\]xdx\\ =2{\mathrm{\πt}}_0{r}_0^2\left(\frac{E^{*}{H}^2}{648}{p}^4+\frac{{\mathrm{\σ}}_{s}H-E^{*}{\mathrm{\ϵ}}_{e}H}{32}{p}^2+\frac{{\mathrm{E\ϵ}}_e^2+E^{*}{\mathrm{\ϵ}}_e^2}{4}-\frac{{\mathrm{\σ}}_s{\mathrm{\ϵ}}_e}{2}\right)\end{array}---\left(20\right)$

More clear succinct for making formula (20) state, carry out variable replacement

U _t＝A ₁p ⁴+A ₂p ²+A ₃(21)

In formula, A ₁, A ₂and A ₃for intermediate variable, can be expressed as

$A_1=\frac{{\mathrm{\πE}}^{*}{t}_0{r}_0^2{H}^2}{324}---\left(22\right)$

$A_2=\frac{{\mathrm{\πt}}_0{r}_0^2({\mathrm{\σ}}_{s}H-E^{*}{\mathrm{\ϵ}}_{e}H)}{16}---\left(23\right)$

$A_3={\mathrm{\πt}}_0{r}_0^2(\frac{{\mathrm{E\ϵ}}_e^2+E^{*}{\mathrm{\ϵ}}_e^2}{2}-{\mathrm{\σ}}_s{\mathrm{\ϵ}}_e)---\left(24\right)$

In like manner can obtain, the strain energy of circular metal thin plate bending distortion

$\begin{array}{c}{U}_b=\underset{V}{\∫}\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s({\mathrm{\ϵ}}_{rb}-{\mathrm{\ϵ}}_e)+\frac{E^{*}{({\mathrm{\ϵ}}_{rb}-{\mathrm{\ϵ}}_e)}^2}{2}\]dV+\underset{V}{\∫}\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s({\mathrm{\ϵ}}_{\mathrm{\θ}b}-{\mathrm{\ϵ}}_e)+\frac{E^{*}{({\mathrm{\ϵ}}_{\mathrm{\θ}b}-{\mathrm{\ϵ}}_e)}^2}{2}\]dV\\ =4{\mathrm{\πr}}_0^2\[\frac{E^{*}{t}_0^3{p}^2}{3072{\mathrm{\π}}^2{D}^2}+\frac{{\mathrm{\σ}}_s{t}_0^{2}p}{32\mathrm{\π}D}(\frac{1}{2}+e^{-2})-\frac{E^{*}{\mathrm{\ϵ}}_e{t}_0^{2}p}{32\mathrm{\π}D}(\frac{1}{2}+e^{-2})+\frac{{\mathrm{Et}}_0{\mathrm{\ϵ}}_e^2+E^{*}{t}_0{\mathrm{\ϵ}}_e^2}{4}-\frac{{\mathrm{\σ}}_s{t}_0{\mathrm{\ϵ}}_e}{2}\\ =A_4{p}^2+A_{5}p+A_6\end{array}---\left(25\right)$

In formula, A ₄, A ₅and A ₆for intermediate variable, can be expressed as

$A_4=\frac{E^{*}{r}_0^2{t}_0^3}{768{\mathrm{\πD}}^2}---\left(26\right)$

$A_5=\frac{{\mathrm{\σ}}_s{r}_0^2{t}_0^2}{8D}(\frac{1}{2}+e^{-2})-\frac{E^{*}{\mathrm{\ϵ}}_e{r}_0^2{t}_0^2}{8D}(\frac{1}{2}+e^{-2})---\left(27\right)$

$A_6={\mathrm{\πt}}_0{r}_0^2({\mathrm{E\ϵ}}_e^2+E^{*}{\mathrm{\ϵ}}_e^2-2{\mathrm{\σ}}_s{\mathrm{\ϵ}}_e)---\left(28\right)$

Circular metal thin plate total strain energy can be expressed as

U＝U _t+U _b＝A ₁p ⁴+(A ₂+A ₄)p ²+A ₅p+(A ₃+A ₆) (29)

The impact energy of alluvium is

Q＝mgh (30)

In formula, m is alluvium quality, and g is acceleration of gravity, and h is shock height.Usual Q characterizes impact energy traditionally.

Total impact energy also needs the impact considering pit depth, namely

Q ^*＝mg(h+δ) (31)

In formula, δ is pit depth.

The impact dent degree of depth can be expressed as

$\mathrm{\δ}={w|}_{r=0}=\frac{{\mathrm{pr}}_0^2}{16\mathrm{\π}D}---\left(32\right)$

According to basic assumption (4) above, because impact energy is all converted into strain energy, namely

Q ^*＝U (33)

$mg(h+\frac{{\mathrm{pr}}_0^2}{16\mathrm{\π}D})=A_1{p}^4+(A_2+A_4)p^2+A_{5}p+A_3+A_6---\left(34\right)$

$A_1{p}^4+(A_2+A_4)p^2+A_{5}p-\frac{{\mathrm{mgr}}_0^2}{16\mathrm{\π}D}p+A_3+A_6-Q=0---\left(35\right)$

A ₁p ⁴+(A ₂+A ₄)p ²+(A ₅-B ₁)p+A ₃+A ₆-Q＝0 (37)

Wherein, B ₁for intermediate variable, can be expressed as

$B_1=\frac{{\mathrm{mgr}}_0^2}{16\mathrm{\π}D}---\left(38\right)$

Be easy to obtain the unknown quantity p in equation (37) by numerical method, then the solution of p is substituted in formula (10) and formula (32), the impact dent distortion corresponding with impact energy Q and impact dent degree of depth δ can be determined.In addition, when alluvium is horizontal impact, when namely pit depth can not cause additional impact energy energy, the B in formula (37) need only be made ₁be 0.

Accompanying drawing explanation

Fig. 1 is the schematic diagram of circular metal thin plate by low velocity impact of clamped constraint.

Fig. 2 is that the circular metal thin plate of clamped constraint is by the stress form simplified during low velocity impact.

Fig. 3 is rigid-plastic linear strain-hardening material stress-strain constitutive relation.

Fig. 4 is be the FB(flow block) of the method for the invention.

In figure, symbol description is as follows:

X in Fig. 1 is the coordinate under rectangular coordinate system, and y is the coordinate under rectangular coordinate system, and z is the range coordinate under rectangular coordinate system, and o is the initial point under rectangular coordinate system, and r is radial distance coordinate under cylindrical-coordinate system, and θ is azimuthal coordinate under cylindrical-coordinate system.

P in Fig. 2 is the centre-point load of circular metal thin plate central role, Q _rfor apart from circular metal thin plate center be the concentric circles place of r face outside shearing.

E in Fig. 3 be in linear strain-hardening material model stress-strain curve in the rate of curve of elastic stage, E ^*for stress-strain curve in linear strain-hardening material model is in the rate of curve of plastic stage, σ is stress, σ _sfor the yield stress of linear strengthening material model, ε is strain, ε _efor maximum elastic strain.

Embodiment

Fig. 4 is the FB(flow block) of the method for the invention, and part 4 step of the present invention realizes, and is specially:

The assumed condition of the new method of step one, proposition prediction circular metal thin plate low velocity impact dimple size.

Assumed condition comprises:

(1) the impact dent shape of sheet metal is rotational symmetric, and does not consider the impact of resilience on dimple size;

(2) meet kirchhoff-Le Fu thin plate hypothesis, therefore can ignore the impact of normal stress and transverse shear stresses outside face, be mainly radial drawing stress, radially bend stress and circumferential skewing stress;

(3) material of sheet metal is elastoplasticity linear strain-hardening material, and as shown in Figure 3, its middle conductor OA is elastic stage to its strain-stress relation, and line segment AB is the plastic stage, and the slope of its correspondence is respectively E and E ^*, ε _efor elastic limit strain;

(4) ignore air resistance, impact process friction force homenergic dissipates, think that impact energy is all converted into strain energy.

Step 2, according to loading during circular metal thin plate low velocity impact and impact dent feature, determine the warping function of impact dent.

According to basic assumption (1) above, because impact dent shape has rotational symmetry, therefore, can describe impact dent shape under cylindrical-coordinate system r θ z, wherein, r is radial distance coordinate, θ azimuthal coordinate.In fig. 1 and 2, in impact dent region, the z in face represents to distortion with w, and known w is only the function about r, and has nothing to do with θ.Circular metal sheet edges is clamped, and central role has centre-point load p, and Ze Ju center is that the concentric circles place of r has total shearing need balance with centre-point load p, specifically can be expressed as

2πrQ _r＝p (1)

In formula, Q _rfor apart from plate center being the shearing at the concentric circles place of r.

Polar coordinates formula according to shearing:

$Q_r=-D\frac{d}{dr}(\frac{d^{2}w}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{dw}{dr})---\left(2\right)$

Formula (2) is substituted in formula (1), can obtain

$-2\mathrm{\π}rD\frac{d}{dr}(\frac{d^{2}w}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{dw}{dr})=p---\left(3\right)$

$-D\frac{d}{dr}(\frac{d^{2}w}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{dw}{dr})=\frac{p}{2\mathrm{\π}r}---\left(4\right)$

To formula (4) integration three times, can obtain

$w=\frac{p}{8\mathrm{\π}D}(r^2\ln r+C_1{r}^2+C_2\ln r+C_3)---\left(5\right)$

In formula, C ₁, C ₂and C ₃for undetermined constant; D is bending stiffness, specifically can be expressed as

$D=\frac{E^{*}{t}_0^3}{12(1-v^2)}---\left(6\right)$

In formula, E ^*for surrender section modulus, v is Poisson ratio, t ₀for the thickness of plate.

Circular metal thin plate clamped constraint time boundary condition need meet:

When r=0 warping function need meet as downstream condition:

${\frac{dw}{dr}|}_{r=0}=f^{\′}\left(0\right)=0---\left(7\right)$

Work as r=r ₀warping function need meet as downstream condition:

$\left\{\begin{array}{c}{w|}_{r=r_0}=f\left(r_0\right)=0\\ {\frac{dw}{dr}|}_{r=r_0}=f^{\′}\left(r_0\right)=0\end{array}\right.---\left(8\right)$

Formula (5) is substituted into respectively in formula (7) and formula (8), can obtain

$\left\{\begin{array}{c}{C}_1=-\frac{1}{2}-{\mathrm{lnr}}_0\\ C_2=0\\ C_3=\frac{1}{2}{r}_0^2\end{array}\right.---\left(9\right)$

Formula (9) is substituted in formula (5), can obtain

$w=\frac{p}{8\mathrm{\π}D}(r^2\ln \frac{r}{r_0}+\frac{r_0^2-r^2}{2})---\left(10\right)$

Step 3, the circular metal thin plate impact dent warping function utilizing step 2 to propose, set up corresponding strain stress relation further.

According to basic assumption (2) above, circular metal thin plate by the deformation process of low velocity impact, mainly by radial-draw deformation, radially bend distortion and circumferential skewing distortion apparatus with shock absorbing.

Circular metal thin plate radial drawing strain stress _rtcan be expressed as

${\mathrm{\ϵ}}_{rt}=\frac{\sqrt{{\mathrm{dw}}^2+{\mathrm{dr}}^2}-dr}{dr}=\sqrt{{\left(\frac{dw}{dr}\right)}^2+1}-1=\frac{1}{2}{\left(\frac{dw}{dr}\right)}^2---\left(11\right)$

Formula (10) is substituted in formula (11), can obtain

${\mathrm{\ϵ}}_{rt}=\frac{p^2}{32{\mathrm{\π}}^2{D}^2}{r}^2{\left(ln\frac{r}{r_0}\right)}^2---\left(12\right)$

Carry out variable replacement, order

$x=\frac{r}{r_0}---\left(13\right)$

The variable of formula (13) is replaced and is equivalent to carry out normalized to variable r, therefore 0≤x≤1.

Formula (13) is substituted in formula (12), can obtain

${\mathrm{\ϵ}}_{rt}=\frac{p^2{r}_0^2}{32{\mathrm{\π}}^2{D}^2}{x}^2{\left(\ln x\right)}^2={\mathrm{Hp}}^2{x}^2{\left(\ln x\right)}^2---\left(14\right)$

In formula, H is constant, can be expressed as

$H=\frac{r_0^2}{32{\mathrm{\π}}^2{D}^2}---\left(15\right)$

Thin plate radially bend curvature κ _rwith circumferential skewing curvature κ _θcan be expressed as

$\left\{\begin{array}{c}{\mathrm{\κ}}_r=\frac{d^{2}w}{{\mathrm{dr}}^2}=\frac{p}{4\mathrm{\π}D}(ln\frac{r}{r_0}+1)\\ {\mathrm{\κ}}_{\mathrm{\θ}}=\frac{1}{r}\left(\frac{dw}{dr}\right)=\frac{p}{4\mathrm{\π}D}\ln \frac{r}{r_0}\end{array}\right.---\left(16\right)$

Formula (13) is substituted in formula (16), can obtain

$\left\{\begin{array}{c}{\mathrm{\κ}}_r=\frac{p}{4\mathrm{\π}D}(\ln x+1)\\ {\mathrm{\κ}}_{\mathrm{\θ}}=\frac{p}{4\mathrm{\π}D}\ln x\end{array}\right.---\left(17\right)$

Thin plate radially bend strain stress _rbwith circumferential skewing strain stress _{θ b}can be expressed as

$\left\{\begin{array}{c}{\mathrm{\ϵ}}_{rb}={\mathrm{\κ}}_{r}z=\frac{p}{4\mathrm{\π}D}z(\ln x+1)\\ {\mathrm{\ϵ}}_{\mathrm{\θ}b}={\mathrm{\κ}}_{\mathrm{\θ}}z=\frac{p}{4\mathrm{\π}D}z\ln x\end{array}\right.---\left(18\right)$

Step 4, according to the principle of work and power, set up circular metal thin plate governing equation under low velocity impact condition, recycling method of value solving solves the undetermined parameter of governing equation, finally determines impact dent size.

According to basic assumption (3) above, the strain energy of circular metal thin plate radial-draw deformation can be expressed as

$U_t=\underset{V}{\∫}(\∫{\mathrm{\σ}}_{rt}d\mathrm{\ϵ})dV=\underset{V}{\∫}\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e)+\frac{E^{*}{({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e)}^2}{2}\]dV---\left(19\right)$

Formula (14) is substituted in formula (19), can obtain

$\begin{array}{c}{U}_t={\∫}_0^{r_0}{\∫}_{-\frac{t_0}{2}}^{\frac{t_0}{2}}{\∫}_0^{2\mathrm{\π}}\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s\left({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e\right)+\frac{E^{*}{\left({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e\right)}^2}{2}\]rd\mathrm{\θ}dzdr\\ =2{\mathrm{\πt}}_0{r}_0^2{\∫}_0^1\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s\left({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e\right)+\frac{E^{*}{\left({\mathrm{\ϵ}}_{rt}-{\mathrm{\ϵ}}_e\right)}^2}{2}\]xdx\\ =2{\mathrm{\πt}}_0{r}_0^2\left(\frac{E^{*}{H}^2}{648}{p}^4+\frac{{\mathrm{\σ}}_{s}H-E^{*}{\mathrm{\ϵ}}_{e}H}{32}{p}^2+\frac{{\mathrm{E\ϵ}}_e^2+E^{*}{\mathrm{\ϵ}}_e^2}{4}-\frac{{\mathrm{\σ}}_s{\mathrm{\ϵ}}_e}{2}\right)\end{array}---\left(20\right)$

More clear succinct for making formula (20) state, carry out variable replacement

U _t＝A ₁p ⁴+A ₂p ²+A ₃(21)

In formula, A ₁, A ₂and A ₃for intermediate variable, can be expressed as

$A_1=\frac{{\mathrm{\πE}}^{*}{t}_0{r}_0^2{H}^2}{324}---\left(22\right)$

$A_2=\frac{{\mathrm{\πt}}_0{r}_0^2({\mathrm{\σ}}_{s}H-E^{*}{\mathrm{\ϵ}}_{e}H)}{16}---\left(23\right)$

$A_3={\mathrm{\πt}}_0{r}_0^2(\frac{{\mathrm{E\ϵ}}_e^2+E^{*}{\mathrm{\ϵ}}_e^2}{2}-{\mathrm{\σ}}_s{\mathrm{\ϵ}}_e)---\left(24\right)$

In like manner can obtain, the strain energy of circular metal thin plate bending distortion

$\begin{array}{c}{U}_b=\underset{V}{\∫}\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s({\mathrm{\ϵ}}_{rb}-{\mathrm{\ϵ}}_e)+\frac{E^{*}{({\mathrm{\ϵ}}_{rb}-{\mathrm{\ϵ}}_e)}^2}{2}\]dV+\underset{V}{\∫}\[\frac{{\mathrm{E\ϵ}}_e^2}{2}+{\mathrm{\σ}}_s({\mathrm{\ϵ}}_{\mathrm{\θ}b}-{\mathrm{\ϵ}}_e)+\frac{E^{*}{({\mathrm{\ϵ}}_{\mathrm{\θ}b}-{\mathrm{\ϵ}}_e)}^2}{2}\]dV\\ =4{\mathrm{\πr}}_0^2\[\frac{E^{*}{t}_0^3{p}^2}{3072{\mathrm{\π}}^2{D}^2}+\frac{{\mathrm{\σ}}_s{t}_0^{2}p}{32\mathrm{\π}D}(\frac{1}{2}+e^{-2})-\frac{E^{*}{\mathrm{\ϵ}}_e{t}_0^{2}p}{32\mathrm{\π}D}(\frac{1}{2}+e^{-2})+\frac{{\mathrm{Et}}_0{\mathrm{\ϵ}}_e^2+E^{*}{t}_0{\mathrm{\ϵ}}_e^2}{4}-\frac{{\mathrm{\σ}}_s{t}_0{\mathrm{\ϵ}}_e}{2}\\ =A_4{p}^2+A_{5}p+A_6\end{array}---\left(25\right)$

In formula, A ₄, A ₅and A ₆for intermediate variable, can be expressed as

$A_4=\frac{E^{*}{r}_0^2{t}_0^3}{768{\mathrm{\πD}}^2}---\left(26\right)$

$A_5=\frac{{\mathrm{\σ}}_s{r}_0^2{t}_0^2}{8D}(\frac{1}{2}+e^{-2})-\frac{E^{*}{\mathrm{\ϵ}}_e{r}_0^2{t}_0^2}{8D}(\frac{1}{2}+e^{-2})---\left(27\right)$

$A_6={\mathrm{\πt}}_0{r}_0^2({\mathrm{E\ϵ}}_e^2+E^{*}{\mathrm{\ϵ}}_e^2-2{\mathrm{\σ}}_s{\mathrm{\ϵ}}_e)---\left(28\right)$

Circular metal thin plate total strain energy can be expressed as

U＝U _t+U _b＝A ₁p ⁴+(A ₂+A ₄)p ²+A ₅p+(A ₃+A ₆) (29)

The impact energy of alluvium is

Q＝mgh (30)

In formula, m is alluvium quality, and g is acceleration of gravity, and h is shock height.Usual Q characterizes impact energy traditionally.

Total impact energy also needs the impact considering pit depth, namely

Q ^*＝mg(h+δ) (31)

In formula, δ is pit depth.

The impact dent degree of depth can be expressed as

$\mathrm{\δ}={w|}_{r=0}=\frac{{\mathrm{pr}}_0^2}{16\mathrm{\π}D}---\left(32\right)$

According to basic assumption (4) above, because impact energy is all converted into strain energy, namely

Q ^*＝U (33)

$mg(h+\frac{{\mathrm{pr}}_0^2}{16\mathrm{\π}D})=A_1{p}^4+(A_2+A_4)p^2+A_{5}p+A_3+A_6---\left(34\right)$

$A_1{p}^4+(A_2+A_4)p^2+A_{5}p-\frac{{\mathrm{mgr}}_0^2}{16\mathrm{\π}D}p+A_3+A_6-Q=0---\left(35\right)$

A ₁p ⁴+(A ₂+A ₄)p ²+(A ₅-B ₁)p+A ₃+A ₆-Q＝0 (37)

Wherein, B ₁for intermediate variable, can be expressed as

$B_1=\frac{{\mathrm{mgr}}_0^2}{16\mathrm{\π}D}---\left(38\right)$

Be easy to obtain the unknown quantity p in equation (37) by numerical method, then the solution of p is substituted in formula (10) and formula (32), the impact dent distortion corresponding with impact energy Q and impact dent degree of depth δ can be determined.In addition, when alluvium is horizontal impact, when namely pit depth can not cause additional impact energy energy, the B in formula (37) need only be made ₁be 0.

Claims (1)

1. predict a method for circular metal thin plate low velocity impact dimple size, it is characterized in that: the method concrete steps are as follows:

The assumed condition of the new method of step one, proposition prediction circular metal thin plate low velocity impact dimple size.

Assumed condition comprises:

(1) the impact dent shape of sheet metal is rotational symmetric, and does not consider the impact of resilience on dimple size;

(2) meet kirchhoff-Le Fu thin plate hypothesis, therefore can ignore the impact of normal stress and transverse shear stresses outside face, be mainly radial drawing stress, radially bend stress and circumferential skewing stress;

(3) material of sheet metal is elastoplasticity linear strain-hardening material, and as shown in Figure 3, elastic stage, line and be the plastic stage, the slope of its correspondence is respectively E and E to its strain-stress relation ^*, ε _efor elastic limit strain;

(4) ignore air resistance, impact process friction force homenergic dissipates, think that impact energy is all converted into strain energy.

Step 2, according to loading during circular metal thin plate low velocity impact and pit feature, determine the warping function of impact dent.

According to basic assumption (1) above, because impact dent shape has rotational symmetry, therefore, can describe impact dent shape under cylindrical-coordinate system r θ z, wherein, r is radial distance coordinate, θ azimuthal coordinate.In impact dent region, the z in face represents to distortion with w, and known w is only the function about r, and has nothing to do with θ.Circular metal sheet edges is clamped, and central role has centre-point load p, and Ze Ju center is that the concentric circles place of r has total shearing need balance with centre-point load p, specifically can be expressed as

2πrQ _r＝p (1)

In formula, Q _rfor apart from plate center being the shearing at the concentric circles place of r.

Polar coordinates formula according to shearing:

Formula (2) is substituted in formula (1), can obtain

To formula (4) integration three times, can obtain

In formula, C ₁, C ₂and C ₃for undetermined constant; D is bending stiffness, specifically can be expressed as

In formula, E ^*for surrender section modulus, v is Poisson ratio, t ₀for the thickness of plate.

Circular metal thin plate clamped constraint time boundary condition need meet:

When r=0 warping function need meet as downstream condition:

Work as r=r ₀warping function need meet as downstream condition:

Formula (5) is substituted into respectively in formula (7) and formula (8), can obtain

Formula (9) is substituted in formula (5), can obtain

Step 3, the circular metal thin plate impact dent warping function utilizing step 2 to propose, set up corresponding strain stress relation further.

According to basic assumption (2) above, circular metal thin plate by the deformation process of low velocity impact, mainly by radial-draw deformation, radially bend distortion and circumferential skewing distortion apparatus with shock absorbing.

Circular metal thin plate radial drawing strain stress _rtcan be expressed as

Formula (10) is substituted in formula (11), can obtain

Carry out variable replacement, order

The variable of formula (13) is replaced and is equivalent to carry out normalized to variable r, therefore 0≤x≤1.

Formula (13) is substituted in formula (12), can obtain

In formula, H is constant, can be expressed as

Thin plate radially bend curvature κ _rwith circumferential skewing curvature κ _θcan be expressed as

Formula (13) is substituted in formula (16), can obtain

Thin plate radially bend strain stress _rbwith circumferential skewing strain stress _{θ b}can be expressed as

Step 4, according to the principle of work and power, set up circular metal thin plate governing equation under low velocity impact condition, recycling method of value solving solves the undetermined parameter of governing equation, finally determines impact dent size.

According to basic assumption (3) above, the strain energy of circular metal thin plate radial-draw deformation can be expressed as

Formula (14) is substituted in formula (19), can obtain

More clear succinct for making formula (20) state, carry out variable replacement

U _t＝A ₁p ⁴+A ₂p ²+A ₃(21)

In formula, A ₁, A ₂and A ₃for intermediate variable, can be expressed as

In like manner can obtain, the strain energy of circular metal thin plate bending distortion

In formula, A ₄, A ₅and A ₆for intermediate variable, can be expressed as

Circular metal thin plate total strain energy can be expressed as

U＝U _t+U _b＝A ₁p ⁴+(A ₂+A ₄)p ²+A ₅p+(A ₃+A ₆)(29)

The impact energy of alluvium is

Q＝mgh (30)

In formula, m is alluvium quality, and g is acceleration of gravity, and h is shock height.Usual Q characterizes impact energy traditionally.

Total impact energy also needs the impact considering pit depth, namely

Q ^*＝mg(h+δ) (31)

In formula, δ is pit depth.

The impact dent degree of depth can be expressed as

According to basic assumption (4) above, because impact energy is all converted into strain energy, namely

Q ^*＝U (33)

A ₁p ⁴+(A ₂+A ₄)p ²+(A ₅-B ₁)p+A ₃+A ₆-Q＝0 (37)

Wherein, B ₁for intermediate variable, can be expressed as

Be easy to obtain the unknown quantity p in equation (37) by numerical method, then the solution of p is substituted in formula (10) and formula (32), the impact dent distortion corresponding with impact energy Q and impact dent degree of depth δ can be determined.In addition, when alluvium is horizontal impact, when namely pit depth can not cause additional impact energy energy, the B in formula (37) need only be made ₁be 0.