The forming deviation, mechanical properties and compression failure of porous structures fabricated by laser melting were analyzed
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摘要: 因多孔结构轻质高强度、力学性能可调节的特点，被广泛用于骨骼医疗、航空航天等领域. 为了探索多孔结构选区激光熔化(Selective Laser Melting, SLM)成形误差与压缩失效性能，以钻石型晶格和六孔开口球形两种多孔结构为例，采用理论预测与试验测试研究SLM制造多孔结构的压缩力学行为，使用ANSYS软件对所研究的多孔结构进行准静态压缩模拟，并对SLM成形的多孔结构进行单轴压缩试验，最后结合仿真和试验，观测和分析它们的变形过程和失效机制. 对比后发现数值设计的多孔结构尺寸与最终制造的结构存在偏差，导致力学性能理论值与试验值存在一定差异，但应力应变场变化规律一致. 试验结果表明，在孔隙率50% ~ 80%时，钻石型晶格结构屈服强度为31.85 ~ 182.13 MPa，弹性模量为1.45 ~ 2.30 GPa；六孔开口球形结构屈服强度为35.19 ~ 130.64 MPa，弹性模量为1.59 ~ 2.90 GPa，不同多孔结构随孔隙率的增大，力学性能变化趋势不一致.Abstract: Due to the characteristics of light, high strength and adjustable mechanical properties of porous structure, it is widely used in bone medicine, aerospace and other fields. In order to explore the forming error and compression failure performance of porous structure with selective laser melting (SLM), this paper takes two kinds of porous structure with diamond lattice and spherical six-hole opening as examples to study the compressive mechanical behavior of porous structure manufactured by SLM by theoretical prediction and experimental test. ANSYS software was used to simulate the quasi-static compression of the studied porous structure, and the uniaxial compression experiment of the SLM formed porous structure was carried out. Finally, the deformation process and failure mechanism of the SLM formed porous structure were observed and analyzed combined with the simulation and experiment. After comparison, it is found that the size of the numerical design porous structure deviates from that of the final manufactured structure, resulting in a certain difference between the theoretical value of mechanical properties and the experimental value, but the variation law of stress and strain field is consistent. The experimental results show that when the porosity is 50% ~ 80%, the yield strength and elastic modulus of diamond lattice structure are 31.85 ~ 182.13 MPa and 1.45 ~ 2.30 GPa respectively. The yield strength and elastic modulus of six-hole spherical structure are 35.19 ~ 130.64 MPa and 1.59 ~ 2.90 GPa respectively. The mechanical properties of different porous structures vary with the increase of porosity.
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0. 序言
合金在焊接过程中，容易产生凝固裂纹缺陷[1]. 通常凝固裂纹发生在固-液两相共存的区域，该区域被称为糊状区. 凝固裂纹的产生与熔池中枝晶的长大过程、显微组织等因素密切相关. Kou[2]指出，枝晶长大所形成的广泛的桥接可以抵抗裂纹，且金属熔体沿晶界(grain boundary, GB)补给越容易，凝固裂纹敏感性越低. 考虑到熔池枝晶的长大行为，提出一个凝固裂纹敏感性指数
$ \left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right| $ ，用来评价合金的凝固裂纹敏感性[2]，其中T为温度，fs为糊状区固体分数. Soysal和Kou[3]用试验成功地验证了该指数在预测合金凝固裂纹敏感性时的有效性.为了更好地理解焊接凝固裂纹行为，必须更仔细地观察熔池枝晶的长大过程. 通过现有的试验手段熔池中枝晶的长大是不可见的. 随着相场(phase field,PF)模拟方法的发展，在不考虑界面跟踪的情况下，PF可以有效地预测合金熔池中枝晶的生长行为[4-5]，研究各因素对凝固过程的影响[6]. 基于PF模拟结果，可以计算获得糊状区的fs值，将其输入到
$\left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right|$ 中，便可以讨论合金的凝固裂纹敏感性. 华中科技大学Geng[7]和南京理工大学Wang等人[8]利用PF方法模拟了铝合金焊接熔池枝晶的生长，并根据模拟结果讨论了凝固裂纹敏感性.合金各向异性影响凝固过程中枝晶的生长，也会影响凝固裂纹的萌生. 采用PF模拟方法模拟Al-Cu合金在不同界面各向异性强度下熔池中枝晶的生长. 根据PF模拟结果，分析枝晶间的桥接、液相通道随各向异性强度的变化规律，进而分析凝固裂纹敏感性. 最后基于PF模拟结果计算fs值，将其导入
$ \left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right| $ 模型中，讨论各向异性强度对凝固裂纹敏感性的影响.1. 模拟方法
应用二维PF模型，模拟Al-4%Cu (质量分数)焊接熔池枝晶的生长. 基于模拟结果，探讨各向异性强度对凝固裂纹敏感性的影响. PF模型中温度场控制方程如为
$$ {T}\left({z},{t}\right)={{T}}_{0}+{G}\left({z}-{R}{t}\right) $$ (1) 式中：
$ {{T}}_{0} $ 为参考温度；z为沿枝晶生长方向；R为凝固速率；G为温度梯度.PF模型的控制方程[9]为
$$\begin{split} {{\tau }}_{0}{a}{\left(\widehat{{n}}\right)}^{2}\left[1-\left(1-{k}\right)\frac{{z}-{R}{t}}{{{l}}_{{{\rm{T}}}}}\right]\frac{\partial {\varphi }}{\partial {t}}={{W}}^{2}\overrightarrow{\nabla }\left[{{a}\left(\widehat{{n}}\right)}^{2}\overrightarrow{\nabla }{\varphi }\right]+ {\varphi }-{{\varphi }}^{3}-{\lambda }{\left(1-{{\varphi }}^{2}\right)}^{2}\left({U}+\frac{{z}-{R}{t}}{{{l}}_{{{\rm{T}}}}}\right) \end{split}$$ (2) $$\begin{split} \left(\frac{1+{k}}{2}-\frac{1-{k}}{2}{\varphi }\right)\frac{\partial {U}}{\partial {t}}=\overrightarrow{\nabla }\Biggr(D\frac{1-{\varphi }}{2}\overrightarrow{\nabla }{U}+\frac{1}{2\sqrt{2}}\left[1+\right. \left.\left. \left(1-{k}\right){U}\right]\frac{\partial {\varphi }}{\partial {t}}\frac{\overrightarrow{\nabla }{\varphi }}{\left|\overrightarrow{\nabla }{\varphi }\right|}\right)+\frac{1}{2}\left[\frac{\partial {\varphi }}{\partial {t}}+\left(1-{k}\right){U}\frac{\partial {\varphi }}{\partial {t}}\right] \end{split}$$ (3) 式中：
$ {{\tau }}_{0} $ 为弛豫时间；$ {\varphi } $ 是相场参量，在固相处为+1，在液相处为−1，在固-液界面处连续变化.$ {U}= $ $ \dfrac{1}{1-{k}}\left(\dfrac{2{k}{c}/{{c}}_\infty }{1-{\varphi }+{k}\left(1+{\varphi }\right)}-1\right) $ 是过饱和场，c是溶质场，$ {{c}}_\infty $ 是初始合金含量；$ {{l}}_{{{\rm{T}}}}=\left|{m}\right|{{c}}_\infty \left(1/k-1\right)/{G} $ 是热长度；m是液相线的斜率；$ {\lambda } $ 为耦合系数；D为扩散系数.模型中各向异性定义为
$$ {{a}\left(\hat{{n}}\right)=\mathrm{\gamma }}_{0}\left[1+{\mathrm{\gamma }}_{4}\mathrm{c}\mathrm{o}\mathrm{s}4\mathrm{\theta }\right] $$ (4) 式中：
$ {\mathrm{\gamma }}_{0} $ 是表面张力的各向同性部分；$ {\mathrm{\gamma }}_{4} $ 是各向异性强度；$ \mathrm{\theta } $ 是界面法线与固定晶轴之间的角度.在PF模型中，Cu元素的反扩散不明显，因此被忽略. 工作采用的固定网格尺寸和时间步长分别为
$\Delta {x}=0.8\;{{{W}}}$ (W为界面厚度)，$ \Delta {t}=0.003{{\tau }}_{0} $ ，其中W = 0.27 µm，$ {{\tau }}_{0}=\;47{\lambda }{{W}}^{2}/75{D} $ . 耦合系数$ {\lambda }=5\sqrt{2}{W}/{8{d}}_{0} $ ，其中${{d}}_{0}= \varGamma /{{l}}_{{{\rm{T}}}}$ 是毛细长度，$\varGamma$ 是Gibbs-Thomson系数. Al-4.0%Cu合金的物理性能参数如表1所示.表 1 Al-4.0%Cu合金的物性参数Table 1. Physical properties of the Al-4.0% Cu alloy扩散系数，D/(10−9 m2·s−1) 液相线斜率m/(K·(wt.%)−1) 溶质分配系数 k Gibbs-Thomson
系数 $ \varGamma $/(10−7 K·m)各向异性强度 $ {\mathrm{\gamma }}_{4} $ 3.0 −2.6 0.48 2.4 0.01 ~ 0.05 如图1所示，计算区域尺寸为518.4 μm × 129.6 μm. 首先在计算区域的底部放置一个平面晶，其高度约为10.8 μm，平面晶粒与液体的界面被认为是熔池中的熔合线. 在计算PF模型时，平面晶沿计算区域长度方向生长. 与铸造相比，焊接时熔池中的冷却速度更快. 因此在PF模型中输入温度梯度G = 150 K/mm，凝固速率R = 1.0 mm/s模拟晶粒长大，使熔池中的冷却速率达到较大的250 K/s.
2. 结果与讨论
2.1 桥接对凝固裂纹敏感性的影响
侧枝的生长可能相互重叠，相邻的柱状晶容易形成桥接，促进刚性固体骨架的形成，从而使应力传递，凝固裂纹不易发生，凝固裂纹敏感性降低[10]. 如图2所示，当
$ {\mathrm{\gamma }}_{4} $ 为0.01, 0.02和0.03时，观察到具有枝晶特征的柱状晶，并且侧枝会发生重叠，形成桥接，具有较低的凝固裂纹敏感性. 当$ {\mathrm{\gamma }}_{4} $ 为0.04和0.05时，观察到具有明显胞状特征的柱状晶，柱状晶之间很难形成桥接，具有较高的凝固裂纹敏感性.![]()
图 2 不同各向异性强度下糊状区晶粒生长和液相通道的模拟结果Figure 2. Simulated grain growth and liquid channels in the mushy zone under different anisotropic strength2.2 液相补给对凝固裂纹敏感性的影响
在
$ {\mathrm{\gamma }}_{4} $ 为0.01，0.02和0.03时，可观察到柱状晶粒之间的液体通道由于出现了侧枝而变得明显起伏. 侧枝生长形成的桥接破坏了沿GB的长液相通道. 不同各向异性强度下的最长液体通道在模拟结果中用黄线标出. 结果表明，随着各向异性强度的增加，最长液相通道的长度也增加. 当液相通道较长时，沿GB进行液相补给较难. 因此在拉应力作用下，各向异性强度越大，液相补给越困难，凝固裂纹敏感性越高.2.3 凝固裂纹敏感性指数
根据PF模拟结果，计算不同各向异性强度下fs随温度T的变化，如图3所示. 值得注意的是，(fs)1/2-T曲线的斜率为
$ \left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right| $ . 计算$ \left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right| $ 判断凝固裂纹敏感性时，需要确定一个特殊点fSB. 当枝晶生长过程出现广泛的桥接时，此时的固相分数fSB被认为是临界值[11]. 因为广泛的桥接会形成坚固的固相骨架，从而可以有效地抑制凝固开裂. 超过fSB时，不能再用$ \left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right| $ 的数值判断凝固裂纹的敏感性. 根据PF模拟结果，当各向异性强度分别为0.01，0.02，0.03，0.04和0.05时，广泛桥接分别出现在(fSB)1/2等于0.93，0.94，0.94，0.95，0.95处. 在特殊点(fSB)1/2处的T-(fSB)1/2曲线中计算斜率$\left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right|$ 可以判断凝固裂纹敏感性. 图4比较了在不同各向异性强度下通过PF模拟结果计算的$\left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right|$ . 结果表明，随着各向异性强度的增加，凝固裂纹敏感性指数$\left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right|$ 也增加，具有较高的凝固裂纹敏感性. 这与上述对柱状晶桥接和液相补给的分析结果一致.3. 结论
(1) 在较大的各向异性强度下，枝晶尖端较稳定，不易形成侧枝，较难形成桥接，表明具有较高的凝固裂纹敏感性.
(2) 在较大的各向异性强度下，在枝晶之间容易形成较长的液相通道，液相补给更困难，表明具有较高的凝固裂纹敏感性.
(3) 基于相场模拟结果，计算的
$ \left|{\rm{d}}T/{\rm{d}}{\left({f}_{{\rm{s}}}\right)}^{1/2}\right| $ 随各向异性强度的增大而增大，表明各向异性强度越大，凝固裂纹敏感性越高.通过目前的试验手段改变合金各向异性几乎是不可能的，因此不能用试验结果验证结论. 但以前的研究成果证明用PF模拟结果，评价合金的凝固裂纹敏感性具有科学性，研究结论可靠.
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图 1 多孔胞元模型
Figure 1. Porous cell model. (a) diamond lattice; (b) spherical six-hole opening structure
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图 5 不同孔隙率钻石型晶格多孔件表面形貌
Figure 5. Surface morphology of diamond lattice porous parts with different porosity. (a) porosity 50%; (b) porosity 60%; (c) porosity 70%; (d) porosity 80%
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图 6 50% ~ 80%孔隙率钻石型晶格结构试验及模拟的应力—应变曲线
Figure 6. Experimental and simulated stress-strain curves of diamond lattice structure with 50% ~ 80% porosity. (a) porosity 50%; (b) porosity 60%; (c) porosity 70%; (d) porosity 80%
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图 7 50% ~ 80%孔隙率六孔开口球形多孔结构试验及模拟的应力—应变曲线
Figure 7. Experimental and simulated stress-strain curves of spherical porous structures with six-hole openings with 50% ~ 80% porosity. (a) porosity 50%; (b) porosity 60%; (c) porosity 70%; (d) porosity 80%
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图 8 孔隙率对多孔结构力学性能的影响
Figure 8. Effect of porosity on mechanical properties of porous structures. (a) change trend of yield strength-porosity; (b) change trend of modulus of elasticity-porosity
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图 9 多孔结构压缩变形过程
Figure 9. Compression deformation process of porous structure. (a) diamond lattice with 80% porosity; (b) spherical six-hole opening structure with 80% porosity
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图 10 压缩仿真(左)和试验(右)各阶段变形
Figure 10. Deformation at different stages of compression simulation (left) and experiment (right). (a) diamond lattice; (b) spherical six-hole opening structure
表 1 多孔结构设计参数
Table 1 Design parameters of porous structure
孔隙率A(%) 钻石型晶格-杆直径d/mm 六孔开口球形-壁厚t/mm 50 1.2706 0.7498 60 1.0918 0.5001 70 0.9106 0.3355 80 0.7162 0.2070 
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表 2 316L粉末的化学成分(质量分数，%)
Table 2 Chemical composition of 316L powder
Si Cr Ni Mn Mo C S P Fe 0.64 16.79 11.07 0.68 2.53 0.027 0.0056 0.022 余量
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表 3 钻石型晶格多孔试样实测参数与设计参数对比
Table 3 Comparison between measured parameters and design parameters of diamond lattice porous sample
孔隙率A(%) 杆件直径d/mm 误差∆d/mm 相对误差B(%) 理论值 实际值 50 1.270 1.311 0.041 3.24 60 1.091 1.166 0.075 6.86 70 0.910 0.976 0.066 7.27 80 0.716 0.789 0.073 10.16
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