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【中文4900字】沖壓變形
沖壓變形工藝可完成多種工序，其基本工序可分為分離工序和變形工序兩 大類。
分離工序是使坯料的一部分與另一部分相互分離的工藝方法，主要有落料、 沖孔、切邊、剖切、修整等。其中有以沖孔、落料應(yīng)用最廣。變形工序是使坯 料的一部分相對(duì)另一部分產(chǎn)生位移而不破裂的工藝方法，主要有拉深、彎曲、 局部成形、脹形、翻邊、縮徑、校形、旋壓等。
從本質(zhì)上看，沖壓成形就是毛坯的變形區(qū)在外力的作用下產(chǎn)生相應(yīng)的塑性 變形，所以變形區(qū)的應(yīng)力狀態(tài)和變形性質(zhì)是決定沖壓成形性質(zhì)的基本因素。因 此，根據(jù)變形區(qū)應(yīng)力狀態(tài)和變形特點(diǎn)進(jìn)行的沖壓成形分類，可以把成形性質(zhì)相 同的成形方法概括成同一個(gè)類型并進(jìn)行系統(tǒng)化的研究。
絕大多數(shù)沖壓成形時(shí)毛坯變形區(qū)均處于平面應(yīng)力狀態(tài)。通常認(rèn)為在板材表面上 不受外力的作用，即使有外力作用，其數(shù)值也是較小的，所以可以認(rèn)為垂直于 板面方向的應(yīng)力為零，使板材毛坯產(chǎn)生塑性變形的是作用于板面方向上相互垂 直的兩個(gè)主應(yīng)力。由于板厚較小，通常都近似地認(rèn)為這兩個(gè)主應(yīng)力在厚度方向 上是均勻分布的?；谶@樣的分析，可以把各種形式?jīng)_壓成形中的毛坯變形區(qū) 的受力狀態(tài)與變形特點(diǎn)，在平面應(yīng)力的應(yīng)力坐標(biāo)系中(沖壓應(yīng)力圖)與相應(yīng)的兩 向應(yīng)變坐標(biāo)系中(沖壓應(yīng)變圖)以應(yīng)力與應(yīng)變坐標(biāo)決定的位置來表示。也就是說， 沖壓應(yīng)力圖與沖壓應(yīng)變圖中的不同位置都代表著不同的受力情況與變形特點(diǎn) (1)沖壓毛坯變形區(qū)受兩向拉應(yīng)力作用時(shí)，可以分為兩種情況：即σ γ >σ >0σ t=0 和σ θ >σ γ >0，σ t=0。再這兩種情況下，絕對(duì)值最大的應(yīng)力都是拉應(yīng)力。以下 對(duì)這兩種情況進(jìn)行分析。
1)當(dāng)σ γ >σ θ >0 且σ t =0 時(shí)，安全量理論可以寫出如下應(yīng)力與應(yīng)變的關(guān)系式：
(1-1) ε γ /（σ γ -σ m）=ε θ /（σ θ -σ m）=ε t/（σ t -σ m）=k
式中 ε γ ，ε θ ，ε t——分別是軸對(duì)稱沖壓成形時(shí)的徑向主應(yīng)變、切向主應(yīng)變 和厚度方向上的主應(yīng)變；
σ γ ，σ θ ，σ t——分別是軸對(duì)稱沖壓成形時(shí)的徑向主應(yīng)力、切向主應(yīng)力和厚度 方向上的主應(yīng)力；
σ m——平均應(yīng)力，σ m=（σ γ +σ θ +σ t）/3；
k——常數(shù)。在平面應(yīng)力狀態(tài)，式（1—1）具有如下形式：
3ε γ /（2σ γ -σ θ ）=3ε θ /（2σ θ -σ t）=3ε t/[-（σ t+σ θ ）]=k （1—2） 因?yàn)棣?γ >σ θ >0，所以必定有 2σ γ -σ θ >0 與ε θ >0。這個(gè)結(jié)果表明：在兩向
拉應(yīng)力的平面應(yīng)力狀態(tài)時(shí)，如果絕對(duì)值最大拉應(yīng)力是σ γ ，則在這個(gè)方向上的主 應(yīng)變一定是正應(yīng)變，即是伸長(zhǎng)變形。
又因?yàn)棣?γ >σ θ >0，所以必定有-（σ t+σ θ ）<0 與ε t<0，即在板料厚度方 向上的應(yīng)變是負(fù)的，即為壓縮變形，厚度變薄。
在σ θ 方向上的變形取決于σ γ 與σ θ 的數(shù)值：當(dāng)σ γ =2σ θ 時(shí)，ε θ =0；當(dāng)σ γ >2
σ θ 時(shí)，ε θ <0；當(dāng) σ γ <2σ θ 時(shí)，ε θ >0。
σ θ 的變化范圍是 σ γ >=σ θ >=0 。在雙向等拉力狀態(tài)時(shí)，σ γ =σ θ ，有 式（1—2）得 ε γ =ε θ >0 及 ε t <0 ；在受單向拉應(yīng)力狀態(tài)時(shí)，σ θ =0，有 式（2—2）可得，ε θ =-ε γ /2。
根據(jù)上面的分析可知，這種變形情況處于沖壓應(yīng)變圖中的 AON 范圍內(nèi)（見 圖 1—1）；而在沖壓應(yīng)力圖中則處于 GOH 范圍內(nèi)（見圖 1—2）。
(1)當(dāng)σ θ >σ γ >0 且σ t=0 時(shí)，有式（1—2）可知：因?yàn)棣?θ >σ γ >0，所以 1）定有 2σ θ >σ γ >0 與ε θ >0。這個(gè)結(jié)果表明：對(duì)于兩向拉應(yīng)力的平面應(yīng)力狀
態(tài)，當(dāng)σ θ 的絕對(duì)值最大時(shí)，則在這個(gè)方向上的應(yīng)變一定時(shí)正的，即一定是 伸長(zhǎng)變形。
又因?yàn)棣?γ >σ θ >0，所以必定有-（σ t+σ θ ）<0 與ε t<0，即在板料厚度方 向上的應(yīng)變是負(fù)的，即為壓縮變形，厚度變薄。
在σ θ 方向上的變形取決于σ γ 與σ θ 的數(shù)值：當(dāng)σ θ =2σ γ 時(shí)，ε γ 0；當(dāng)σ θ >
σ γ ，ε γ <0；當(dāng) σ θ <2σ γ 時(shí)，ε γ >0。
σ γ 的變化范圍是 σ θ >= σ γ >=0 。當(dāng)σ γ =σ θ 時(shí)，ε γ =ε θ >0，也就是 在雙向等拉力狀態(tài)下，在兩個(gè)拉應(yīng)力方向上產(chǎn)生數(shù)值相同的伸長(zhǎng)變形；在受單 向拉應(yīng)力狀態(tài)時(shí)，當(dāng)σ γ =0 時(shí)，ε γ =-ε θ /2，也就是說，在受單向拉應(yīng)力狀態(tài) 下其變形性質(zhì)與一般的簡(jiǎn)單拉伸是完全一樣的。
這種變形與受力情況，處于沖壓應(yīng)變圖中的 AOC 范圍內(nèi)（見圖 1—1）；而 在沖壓應(yīng)力圖中則處于 AOH 范圍內(nèi)（見圖 1—2）。
上述兩種沖壓情況，僅在最大應(yīng)力的方向上不同，而兩個(gè)應(yīng)力的性質(zhì)以及 它們引起的變形都是一樣的。因此，對(duì)于各向同性的均質(zhì)材料，這兩種變形是 完全相同的。
(1)沖壓毛坯變形區(qū)受兩向壓應(yīng)力的作用，這種變形也分兩種情況分析，即
o γ <σ θ <
σ t=0 和σ θ <σ γ <0，σ t=0。
1）當(dāng)σ γ <σ θ <0 且σ t=0 時(shí)，有式（1—2）可知：因?yàn)棣? γ <σ θ <0，一定有
2σ γ -σ θ <0 與ε γ <0。這個(gè)結(jié)果表明：在兩向壓應(yīng)力的平面應(yīng)力狀態(tài)時(shí)，如果
11
絕對(duì)值最大拉應(yīng)力是σ γ <0，則在這個(gè)方向上的主應(yīng)變一定是負(fù)應(yīng)變，即是壓 縮變形。
又因?yàn)棣? γ <σ θ <0，所以必定有-（σ t+σ θ ）>0 與ε t>0，即在板料厚度方 向上的應(yīng)變是正的，板料增厚。
在σ θ 方向上的變形取決于σ γ 與σ θ 的數(shù)值：當(dāng)σ γ =2σ θ 時(shí)，ε θ =0；當(dāng)σ γ >2
σ θ 時(shí)，ε θ <0；當(dāng) σ γ <2σ θ 時(shí)，ε θ >0。
這時(shí)σ θ 的變化范圍是 σ γ 與 0 之間 。當(dāng)σ γ =σ θ 時(shí)，是雙向等壓力狀態(tài) 時(shí)，故有 ε γ =ε θ <0；當(dāng)σ θ =0 時(shí)，是受單向壓應(yīng)力狀態(tài)，所以ε θ =-ε γ /2。 這種變形情況處于沖壓應(yīng)變圖中的 EOG 范圍內(nèi)（見圖 1—1）；而在沖壓應(yīng)力圖 中則處于 COD 范圍內(nèi)（見圖 1—2）。
2) 當(dāng)σ θ <σ γ <0 且σ t=0 時(shí)，有式（1—2）可知：因?yàn)棣? θ <σ γ <0，所以 一定有 2σ θ σ γ <0 與ε θ <0。這個(gè)結(jié)果表明：對(duì)于兩向壓應(yīng)力的平面應(yīng)力狀 態(tài)，如果絕對(duì)值最大是σ θ ，則在這個(gè)方向上的應(yīng)變一定時(shí)負(fù)的，即一定是壓 縮變形。
又因?yàn)棣? γ <σ θ <0，所以必定有-（σ t+σ θ ）>0 與ε t>0，即在板料厚度方 向上的應(yīng)變是正的，即為壓縮變形，板厚增大。
在σ θ 方向上的變形取決于σ γ 與σ θ 的數(shù)值：當(dāng)σ θ =2σ γ 時(shí)，ε γ =0；當(dāng)σ θ >2
σ γ ，ε γ <0；當(dāng) σ θ <2σ γ 時(shí)，ε γ >0。
這時(shí)，σ γ 的數(shù)值只能在σ θ <= σ γ <=0 之間變化。當(dāng)σ γ =σ θ 時(shí)，是雙向 等壓力狀態(tài)，所以ε γ =ε θ <0；當(dāng)σ γ =0 時(shí)，是受單向壓應(yīng)力狀態(tài)，所以有ε γ
=-ε θ /2>0。這種變形與受力情況，處于沖壓應(yīng)變圖中的 GOL 范圍內(nèi)（見圖 1
—1）；而在沖壓應(yīng)力圖中則處于 DOE 范圍內(nèi)（見圖 1—2）。
(1)沖壓毛坯變形區(qū)受兩個(gè)異號(hào)應(yīng)力的作用，而且拉應(yīng)力的絕對(duì)值大于壓應(yīng) 力的絕對(duì)
值。這種變形共有兩種情況，分別作如下分析。
1）當(dāng)σ γ >0，σ θ <0 及|σ γ |>|σ θ |時(shí)，由式（1—2）可知：因?yàn)棣?γ >0，σ θ
<0 及|σ γ |>|σ θ |，所以一定有 2σ γ -σ θ >0 及ε γ >0。這個(gè)結(jié)果表明：在異號(hào)的 平面應(yīng)力狀態(tài)時(shí)，如果絕對(duì)值最大應(yīng)力是拉應(yīng)力，則在這個(gè)絕對(duì)值最大的拉應(yīng) 力方向上應(yīng)變一定是正應(yīng)變，即是伸長(zhǎng)變形。
又因?yàn)棣?γ >0，σ θ <0 及|σ γ |>|σ θ |，所以必定有ε θ <0，即在板料厚度方向 上的應(yīng)變是負(fù)的，是壓縮變形。
這時(shí)σ θ 的變化范圍只能在σ θ =-σ γ 與σ θ =0 的范圍內(nèi) 。當(dāng)σ θ =-σ γ 時(shí)，
ε γ >0ε θ <0 且|ε γ |=|ε θ |；當(dāng)σ θ =0 時(shí)，ε γ >0，ε θ <0，而且ε θ =-ε γ /2，這是
受單向拉的應(yīng)力狀態(tài)。這種變形情況處于沖壓應(yīng)變圖中的 MON 范圍內(nèi)（見圖
1—1）；而在沖壓應(yīng)力圖中則處于 FOG 范圍內(nèi)（見圖 1—2）。
2）當(dāng)σ θ >0，σ γ <0，σ t=0 及|σ θ |>|σ γ |時(shí)，由式（1—2）可知：用與前 項(xiàng)相同的方法分析可得ε θ >0。即在異號(hào)應(yīng)力作用的平面應(yīng)力狀態(tài)下，如果絕 對(duì)值最大應(yīng)力是拉應(yīng)力σ θ ，則在這個(gè)方向上的應(yīng)變是正的，是伸長(zhǎng)變形；而在 壓應(yīng)力σ γ 方向上的應(yīng)變是負(fù)的（ε γ <=0），是壓縮變形。
這時(shí)σ γ 的變化范圍只能在σ γ =-σ θ 與σ γ =0 的范圍內(nèi) 。當(dāng)σ γ =-σ θ 時(shí)， ε θ >0，ε γ <0 且|ε γ |=|ε θ |；當(dāng)σ γ =0 時(shí)，ε θ >0，ε γ <0，而且ε γ =-ε θ /2。 這種變形情況處于沖壓應(yīng)變圖中的 COD 范圍內(nèi)（見圖 1—1）；而在沖壓應(yīng)力圖 中則處于 AOB 范圍內(nèi)（見圖 1—2）。
雖然這兩種情況的表示方法不同，但從變形的本質(zhì)看是一樣的。
(1)沖壓毛坯變形區(qū)受兩個(gè)方向上的異號(hào)應(yīng)力的作用，而且壓應(yīng)力的絕對(duì)值 大于拉應(yīng)力
的絕對(duì)值。以下對(duì)這種變形的兩種情況分別進(jìn)行分析。
1）當(dāng)σ γ >0，σ θ <0 而且|σ θ |>|σ γ |時(shí)，由式（1—2）可知：因?yàn)棣? γ >0， σ θ <0 及|σ θ |>|σ γ |，所以一定有 2σ θ - σ γ <0 及ε θ <0。這個(gè)結(jié)果表明：在異 號(hào)的平面應(yīng)力狀態(tài)時(shí)，如果絕對(duì)值最大應(yīng)力是壓應(yīng)力σ θ ，則在這個(gè)方向上應(yīng)變 是負(fù)的，即是壓縮變形。
又因?yàn)棣?γ >0，σ θ <0，必定有 2σ γ - σ θ <0 及ε γ >0，即在拉應(yīng)力方向上 的應(yīng)變是正的，是伸長(zhǎng)變形。
這時(shí)σ γ 的變化范圍只能在σ γ =-σ θ 與σ γ =0 的范圍內(nèi) 。當(dāng)σ γ =-σ θ 時(shí)， ε γ >0ε θ <0 且ε γ =-ε θ ；當(dāng)σ γ =0 時(shí)，ε γ >0，ε θ <0，而且ε γ =-ε θ /2。這種 變形情況處于沖壓應(yīng)變圖中的 DOF 范圍內(nèi)（見圖 1—1）；而在沖壓應(yīng)力圖中則 處于 BOC 范圍內(nèi)（見圖 1—2）。
2）當(dāng)σ θ >0，σ γ <0，σ t=0 及|σ γ |>|σ θ |時(shí)，由式（1—2）可知：用與前 項(xiàng)相同的方法分析可得ε γ <σ γ 0。即在異號(hào)應(yīng)力作用的平面應(yīng)力狀態(tài)下，如果 絕對(duì)值最大應(yīng)力是壓應(yīng)力σ γ ，則在這個(gè)方向上的應(yīng)變是負(fù)的，是壓縮變形；而 在拉應(yīng)力σ θ 方向上的應(yīng)變是正的，是伸長(zhǎng)變形。
這時(shí)σ θ 的數(shù)值只能介于σ θ =-σ γ 與σ θ =0 的范圍內(nèi) 。當(dāng)σ θ =-σ γ 時(shí)，ε
θ >0，ε γ <0 且ε θ =-ε γ ；當(dāng)σ θ =0 時(shí)，ε θ >0，ε γ <0，而且ε θ =-ε γ /2。這 種變形情況處于沖壓應(yīng)變圖中的 DOE 范圍內(nèi)（見圖 1—1）；而在沖壓應(yīng)力圖中 則處于 BOC 范圍內(nèi)（見圖 1—2）。
這四種變形與相應(yīng)的沖壓成形方法之間是相對(duì)的，它們之間的對(duì)應(yīng)關(guān)系，
用文字標(biāo)注在圖 1—1 與圖 1—2 上。
上述分析的四種變形情況，相當(dāng)于所有的平面應(yīng)力狀態(tài)，也就是說這四種 變形情況可以把全部的沖壓變形毫無遺漏地概括為兩大類別，即伸長(zhǎng)類與壓縮 類。
當(dāng)作用于沖壓毛坯變形區(qū)內(nèi)的拉應(yīng)力的絕對(duì)值最大時(shí)，在這個(gè)方向上的變 形一定是伸長(zhǎng)變形，稱這種變形為伸長(zhǎng)類變形。根據(jù)上述分析，伸長(zhǎng)類變形在 沖壓應(yīng)變圖中占有五個(gè)區(qū)間，即 MON、AON、AOB、BOC 及 COD；而在沖 壓應(yīng)力圖中則占有四個(gè)區(qū)間 FOG、GOH、AOH 及 AOB。
當(dāng)作用于沖壓毛坯變形區(qū)內(nèi)的壓應(yīng)力的絕對(duì)值最大時(shí)，在這個(gè)方向上的變 形一定是壓縮變形，稱這種變形為壓縮類變形。根據(jù)上述分析，壓縮類變形在 沖壓應(yīng)變圖中占有五個(gè)區(qū)間，即 LOM、HOL、GOH、FOG 與 DOF；而在沖壓 應(yīng)力圖中則占有四個(gè)區(qū)間 EOF、DOE、COD、BOC。
MD 與 FB 分別是沖壓應(yīng)變圖與沖壓應(yīng)力圖中兩類變形的分界線。分界線 的右上方是伸長(zhǎng)類變形，而分界線的左下方是壓縮變形。
由于塑性變形過程中材料所受的應(yīng)力和由此應(yīng)力所引起的應(yīng)變之間存在著 相互對(duì)應(yīng)的關(guān)系，所以沖壓應(yīng)力圖與沖壓應(yīng)變圖也一定存在著一定的對(duì)應(yīng)關(guān)系。
每一個(gè)沖壓變形都可以在沖壓應(yīng)力圖上和沖壓應(yīng)變圖上找到它固定的位置。根
據(jù)沖壓毛坯變形區(qū)內(nèi)的應(yīng)力狀態(tài)或變形情況，利用沖壓變形圖或沖壓應(yīng)力圖中 的分界線（MD 或 FB）就可以容易地判斷該沖壓變形的性質(zhì)與特點(diǎn)。
概括以上分析結(jié)果，把各種應(yīng)力狀態(tài)在沖壓應(yīng)變圖和沖壓應(yīng)力圖中所處的 位置以及兩個(gè)圖的對(duì)應(yīng)關(guān)系列于表 1—1。從表 1—1 中的關(guān)系可知，沖壓應(yīng)力
圖與沖壓應(yīng)變圖中各區(qū)間所處的幾何位置并不一樣，但它們?cè)趦蓚€(gè)圖中的順序 是相同的。最重要是一點(diǎn)是：伸長(zhǎng)類與壓縮類變形的分界線，在兩個(gè)圖里都是
與坐標(biāo)軸成 45°角的一條斜線。表 1—2 中列出了伸長(zhǎng)類變形與壓縮類變形在 沖壓成形工藝方面的特點(diǎn)。
從表 1—2 可以清楚地看出，由于每一類別的沖壓成形方法，其毛坯變形區(qū)
的受力與變形特點(diǎn)相同，而與變形有關(guān)的一些規(guī)律也都是一樣的，所以有可能 在對(duì)各種具體的沖壓成形方法進(jìn)行研究之外，開展綜合性的體系化研究工作。 體系化研究方法的特點(diǎn)是對(duì)每一類別沖壓成形方法的共性規(guī)律進(jìn)行研究工作， 體系化研究的結(jié)果對(duì)每一個(gè)屬于該類別的成形方法都是適用的。這種體系化的 研究工作，在板材沖壓性能、沖壓成形極限等方面，已有一定程度的開展。應(yīng) 用體系化方法研究沖壓成形極限的內(nèi)容可用圖 1—3 予以說明。
表 1—1 沖壓應(yīng)力狀態(tài)與沖壓變形狀態(tài)的對(duì)照
應(yīng)力狀態(tài)
沖壓應(yīng)變 圖中位置
沖壓應(yīng)變 圖中位置
在絕對(duì)值最大
的應(yīng)力方向上
變形 類別
應(yīng)力 應(yīng)變
雙向受拉
o θ >0,σ γ >0
o γ > σ
θ
AON
GOH
+ +
伸長(zhǎng)類
o θ >σ γ
AOC
AOH
+ +
伸長(zhǎng)類
雙向受壓
o θ <0,σ γ <0
o γ < σ
θ
EOG
COD
— —
壓縮類
o θ <σ γ
GOL
DOE
— —
壓縮類
異號(hào)應(yīng)力
o γ >0,σ θ <0
|σ γ |>|σ
θ |
MON
FOG
+ +
伸長(zhǎng)類
|σ θ |>|σ
γ |
LOM
EOF
— —
壓縮類
異號(hào)應(yīng)力
o θ >0,σ γ <0
|σ θ |>|σ
γ |
COD
AOB
+ +
伸長(zhǎng)類
|σ γ |> |σ
θ |
DOE
BOC
— —
壓縮類
表 1—2 伸長(zhǎng)類成形與壓縮類成形的對(duì)比
項(xiàng)目
伸長(zhǎng)類成形
壓縮類成形
變形區(qū)質(zhì)量問題的表
現(xiàn)形式
變形程度過大引起變形區(qū)
產(chǎn)生破裂現(xiàn)象
壓力作用下失穩(wěn)起皺
成形極限
1．主要取決于板材的塑
性，與厚度無關(guān)
2．可用伸長(zhǎng)率及成形極
限 DLF 判斷
1．主要取決于傳力區(qū)的
承載能力
2．取決于抗失穩(wěn)能力
3．與板厚有關(guān)
變形區(qū)板厚的變化
減薄
增厚
提高成形極限的方法
1．改善板材塑性
2．使變形均勻化，降低局
部變形程度
3．工序間熱處理
1．采用多道工序成形
2．改變傳力區(qū)與變形區(qū)
的力學(xué)關(guān)系
3．采用防起皺措施
+ε γ
+σ γ
-σ θ
θ +σ
擴(kuò)口
-ε γ
-σ γ
圖 1—3 沖壓應(yīng)變圖
圖 1—3 體系化研究方法舉例
Categories of stamping forming
Many deformation processes can be done by stamping, the basic processes of the stamping can be divided into two kinds: cutting and forming.
Cutting is a shearing process that one part of the blank is cut form the other .It mainly includes blanking, punching, trimming, parting and shaving, where punching and blanking are the most widely used. Forming is a process that one part of the blank has some displacement form the other. It mainly includes deep drawing, bending, local forming, bulging, flanging, necking, sizing and spinning.
In substance, stamping forming is such that the plastic deformation occurs in the deformation zone of the stamping blank caused by the external force. The stress state and deformation characteristic of the deformation zone are the basic factors to decide the properties of the stamping forming. Based on the stress state and deformation characteristics of the deformation zone, the forming methods can be divided into several categories with the same forming properties and to be studied systematically.
The deformation zone in almost all types of stamping forming is in the plane stress state. Usually there is no force or only small force applied on the blank surface. When it is assumed that the stress perpendicular to the blank surface equal to zero, two principal stresses perpendicular to each other and act on the blank surface produce the plastic deformation of the material. Due to the small thickness of the blank, it is assumed approximately that the two principal stresses distribute uniformly along the thickness direction. Based on this analysis, the stress state and
the deformation characteristics of the deformation zone in all kind of stamping forming can be denoted by the point in the coordinates of the plane princ ipal stress(diagram of the stamping stress) and the coordinates of the corresponding plane principal stains (diagram of the stamping strain). The different points in the figures of the stamping stress and strain possess different stress state and deformation characteristics.
(1) When the deformation zone of the stamping blank is subjected toplanetensile stresses, it can be divided into two cases, that is σγ>σθ>0,σt=0andσθ>σγ >0,σt=0.In both cases, the stress with the maximum absolute value is always a tensile stress. These two cases are analyzed respectively as follows.
2)In the case that σγ>σθ>0andσt=0, according to the integral theory, the relationships between stresses and strains are:
εγ/（σγ-σm）=εθ/（σθ-σm）=εt/（σt -σm）=k 1.1
where, εγ，εθ，εt are the principal strains of the radial, tangential and thickness directions of the axial symmetrical stamping forming; σγ，σθand σtare the principal stresses of the radial, tangential and thickness directions of the axial symmetrical stamping forming;σm is the average stress,σm=（σγ+σθ+σt）/3; k is a constant.
In plane stress state, Equation 1.1
3εγ/（2σγ-σθ）=3εθ/（2σθ-σt）=3εt/[-（σt+σθ）]=k 1.2
Since σγ>σθ>0,so 2σγ-σθ>0 and εθ>0.It indicates that in plane stress state with two axial tensile stresses, if the tensile stress with the maximum absolute value is σγ, the principal strain in this direction must be positive, that is, the deformation belongs
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to tensile forming.
In addition, because σγ>σθ>0，therefore -（σt+σθ）<0 and εt<0. The strain in the thickness direction of the blankεt is negative, that is, the deformation belongs to compressive forming, and the thickness decreases.
The deformation condition in the tangential direction depends on the values ofσγ and σθ. When σγ=2σθ,εθ=0； when σγ>2σθ,εθ<0；and when σγ<2σθ ,εθ>0.
The range of σθ is σγ>=σθ>=0 . In the equibiaxial tensile stress state σγ=σθ ， according to Equation 1.2,εγ=εθ>0 and εt <0 . In the uniaxial tensile stress stateσθ=0，according to Equation 1.2 εθ=-εγ/2.
According to above analysis, it is known that this kind of deformation condition is in the region AON of the diagram of the diagram of the stamping strain (see Fig .1.1), and in the region GOH of the diagram of the stamping stress (see Fig.1.2).
2)When σθ>σγ >0 and σt=0, according to Equation 1.2 , 2σθ>σγ >0 and εθ>0,This result shows that for the plane stress state with two tensile stresses, when the absoluste value of σθ is the strain in this direction must be positive, that is, it must be in the state of tensile forming.
Also becauseσγ>σθ>0，therefore -（σt+σθ）<0 and εt<0. The strain in the
thickness direction of the blankεt is negative, or in the state of compressive forming, and the thickness decreases.
The deformation condition in the radial direction depends on the values ofσγ
and σθ. When σθ=2σγ,εγ0;when σθ>σγ,εγ<0;and when σθ<2σγ,εγ>0.
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The range of σγ is σθ>= σγ>=0 .When σγ=σθ,εγ=εθ>0, that is, in equibiaxial tensile stress state, the tensile deformation with the same values occurs in the two tensile stress directions; when σγ=0, εγ=-εθ /2, that is, in uniaxial tensile stress state, the deformation characteristic in this case is the same as that of the ordinary uniaxial tensile.
This kind of deformation is in the region AON of the diagram of the stamping strain (see Fig.1.1), and in the region GOH of the diagram of the stamping stress (see Fig.1.2).
Between above two cases of stamping deformation, the properties ofσθandσγ, and the deformation caused by them are the same, only the direction of the maximum stress is different. These two deformations are same for isotropic homogeneous material.
(1) When the deformation zone of stamping blank is subjected to two compressive stressesσγandσθ(σt=0), it can also be divided into two cases, which are σγ<σθ<0,σt=0 and σθ<σγ <0,σt=0.
1）When σγ<σθ<0 and σt=0, according to Equation 1.2, 2σγ-σθ<0 與 εγ=0.This
result shows that in the plane stress state with two compressive stresses, if the stress with the maximum absolute value is σγ<0, the strain in this direction must be negative, that is, in the state of compressive forming.
Also because σγ<σθ<0, therefore -（σt +σθ）>0 and εt>0.The strain in the thickness
direction of the blankεt is positive, and the thickness increases.
The deformation condition in the tangential direction depends on the values
ofσγ and σθ.When σγ=2σθ,εθ=0;when σγ>2σθ,εθ<0;and when σγ<2σθ ,εθ>0.
The range of σθ is σγ<σθ<0.When σγ=σθ,it is in equibiaxial tensile stress state, henceεγ=εθ<0; when σθ=0,it is in uniaxial tensile stress state, hence εθ=-εγ/2.This kind of deformation condition is in the region EOG of the diagram of the stamping strain (see Fig.1.1), and in the region COD of the diagram of the stamping stress (see Fig.1.2).
2）When σθ<σγ <0and σt=0, according to Equation 1.2,2σθ-σγ <0 and εθ<0. This
result shows that in the plane stress state with two compressive stresses, if the stress with the maximum absolute value is σθ, the strain in this direction must be negative, that is, in the state of compressive forming.
Also becauseσθ<σγ <0 , therefore -（σt +σθ）>0 and εt>0.The strain in the
thickness direction of the blankεt is positive, and the thickness increases.
The deformation condition in the radial direction depends on the values ofσγ and σθ. When σθ=2σγ, εγ=0; when σθ>2σγ,εγ<0; and when σθ<2σγ ,εγ>0.
The range of σγ is σθ<= σγ<=0 . When σγ=σθ , it is in equibiaxial tensile stress state, hence εγ=εθ<0; when σγ=0, it is in uniaxial tensile stress state, hence εγ=-εθ
/2>0.This kind of deformation is in the region GOL of the diagram of the stamping strain (see Fig.1.1), and in the region DOE of the diagram of the stamping stress (see Fig.1.2).
(3) The deformation zone of the stamping blank is subjected to two stresses with opposite signs, and the absolute value of the tensile stress is larger than that of the compressive stress. There exist two cases to be analyzed as follow:
1) When σγ>0, σθ<0 and |σγ|>|σθ|, according to Equation 1.2, 2σγ-σθ>0 and εγ>0.This result shows that in the plane stress state with opposite signs, if the stress with the maximum absolute value is tensile, the strain in the maximum stress direction is positive, that is, in the state of tensile forming.
Also because σγ>0, σθ<0 and |σγ|>|σθ|, therefore εθ<0. The strain in the compressive stress direction is negative, that is, in the state of compressive forming.
The range of σθ is 0>=σθ>=-σγ. When σθ=-σγ, then εγ>0,εθ<0 , and |εγ|=|εθ|;when σθ=0, then εγ>0,εθ<0, and εθ=-εγ/2, it is the uniaxial tensile stress state. This kind of deformation condition is in the region MON of the diagram of the stamping strain (see Fig.1.1), and in the region FOG of the diagram of the stamping stress (see Fig.1.2).
2) When σθ>0, σγ <0,σt=0 and |σθ|>|σγ|, according to Equation 1.2, by
means of the same analysis mentioned above, εθ>0, that is, the deformation zone is in the plane stress state with opposite signs. If the stress with the maximum absolute value is tensile stress σθ, the strain in this direction is positive, that is, in the state of tensile forming. The strain in the radial direction is negative （εγ<=0）, that is, in the state of compressive forming.
The range of σγ is 0>=σγ>=-σθ. When σγ=-σθ, then εθ>0,εγ <0 and |εγ|=|εθ|; when σγ=0, then εθ>0,εγ <0, andεγ=-εθ /2. This kind of deformation condition is in the region COD of the diagram of the stamping strain (see Fig.1.1), and in the region AOB of the diagram of the stamping stress (see Fig.1.2).
Although the expressions of these two cases are different, their deformation
essences are the same.
(4) The deformation zone of the stamping blank is subjected to two stresses with opposite signs, and the absolute value of the compressive stress is larger than that of the tensile stress. There exist two cases to be analyzed as follows:
1) When σγ>0,σθ<0 and |σθ|>|σγ|, according to Equation 1.2, 2σθ- σγ<0 and εθ<0.This result shows that in plane stress state with opposite signs, if the stress with the maximum absolute value is compressive stress σθ, the strain in this direction is negative, or in the state of compressive forming.
Also because σγ>0 and σθ<0, therefore 2σγ- σθ<0 and εγ>0. The strain in the tensile stress direction is positive, or in the state of tensile forming.
The range of σγ is 0>=σγ>=-σθ.When σγ=-σθ, then εγ>0,εθ<0, and εγ=-εθ;when σγ=0, then εγ>0,εθ<0, and εγ=-εθ/2. This kind of deformation is in the region LOM of the diagram of the stamping strain (see Fig.1.1), and in the region EOF of the diagram of the stamping stress (see Fig.1.2).
2) When σθ>0, σγ <0 and |σγ|>|σθ|, according to Equation 1.2 and by means of the same analysis mentioned above,εγ< 0.This result shows that in plane stress state with opposite signs, if the stress with the maximum absolute value is compressive stress σγ,the strain in this direction is negative, or in the state of compressive forming, The strain in the tensile stress direction is positive, or in the state of tensile forming.
The range of σθ is 0>=σθ>=-σγ.When σθ=-σγ, then εθ>0,εγ <0, and εθ=-εγ;when σθ=0, then εθ>0,εγ <0, and εθ=-εγ/2. Such deformation is in the region DOF of the
15
diagram of the stamping strain (see Fig.1.1), and in the region BOC of the diagram of the stamping stress (see Fig.1.2).
The four deformation conditions are related to the corresponding stamping forming methods. Their relationships are labeled with letters in Fig.1.1 and Fig.1.2.
The four deformation conditions analyzed above are applicable to all kinds of plane stress states, that is, the four deformation conditions can sum up all kinds of stamping forming in to two types, tensile and compressive. When the stress with the maximum absolute value in the deformation zone of the stamping blank is tensile, the deformation along this stress direction must be tensile. Such stamping deformation is called tensile forming. Based on above analysis, the tensile forming occupies five regions MON, AON, AOB, BOC and COD in the diagram of the stamping stain; and four regions FOG, GOH, AOH and AOB in the diagram of the stamping stress.
When the stress with the maximum absolute value in the deformation zone of the stamping blank is compressive, the deformation along this stress direction must be compressive. Such stamping deformation is called compressive forming. Based on above analysis, the compressive forming occupies five regions LOM, HOL, GOH, FOG and DOF in the diagram of the stamping strain; and four regions EOF, DOE, COD and BOC in the diagram of the stamping stress.
MD and FB are the boundaries of the two types of forming in the diagrams of the stamping strain and stress respectively. The tensile forming is located in the top right of the boundary, and the comp