基于有限元法探究实验条件对一种耐热钢压痕松弛行为的影响

图1 圆锥形和四棱锥形压头的示意图

Fig.1 Schematics of conical (a) and quadrangular pyramid (b) indenters

采用自主研制的DZY.2000电子式蠕变试验机开展压痕松弛蠕变实验。高温环境由三区独立控温电阻炉(轴向长度500 mm)提供，各温区通过S型热电偶实现闭环控制，通过两组独立调节的加热体实现轴向300 mm范围内温度场的均匀分布，结合缓冲温区的热耦合设计有效抑制了边界热传导引起的非稳态温度波动(波动控制在±1 ℃以内)；力学加载单元集成伺服电动缸驱动系统(最大载荷2 kN)和滚珠丝杠传动机构，通过模块化接口实现恒载荷/恒应变/恒位移多模式加载功能；压头固定在中心控制位移支架上，正对载物台放置试样，确保试样位于压头下方的平台中心。实验在充N₂气氛的高温炉中进行，以抑制试样氧化。升温前持续通入N₂，保温前5~10 min停止通气，避免气流扰动引起温度变化。将高温炉加热至实验温度(750 ℃)，待试样表面温度达到预设值后，保温2 h使炉内温度分布均匀。随后施加20 N初始载荷，确保压头与试样表面稳定接触。以0.5 μm/s速率将压头压入至0.1 mm深度后保持压深恒定，通过高精度位移传感器(精度0.1 μm)实时监测载荷变化，并自动记录载荷、时间、位移及温度数据。试验机负载能力为0~5000 N，数据由计算机同步采集。最后以0.5 μm/s相同速率卸载，完成单次压痕松弛实验。

本工作在有限元模拟中基于两种压头的轴对称特性，采用1/4对称模型进行计算，采用10节点二次四面体单元(C3D10)对试件进行建模。为了准确得到接触区域的应力集中效应并兼顾计算效率，采用非均匀网格划分，在压头尖端及接触区域实施局部网格加密(最小单元尺寸0.005 mm)，网格尺寸沿远离尖端方向呈梯度递增，直至外围区域达到最大单元尺寸(0.3 mm)，试件三维剖面和表面的有限元计算网格如图2所示。模型底部约束完全固定，侧面设置轴对称约束。

图2

图2 试件三维剖面和表面的有限元计算网格

Fig.2 Finite element calculation mesh for the three-dimensional cross-section (a) and surface (b) of the specimen (Insets are corresponding partially enlarged views)

通过拟合Sanicro25钢在圆柱压头下的压痕蠕变实验结果并参考相关文献[23,24]，得到750 ℃条件下模型材料的弹性模量E = 141 GPa，屈服应力σ_y = 74.3 MPa，蠕变应力指数n = 1.85，Poisson比ν = 0.3。由于压头的弹性模量远高于试件，因此不关注压头的变形，在计算中将压头简化为刚体。

2 实验结果

图3展示了圆锥形压头(图3a)和四棱锥形压头(图3b)在最大压痕深度0.1 mm条件下，Sanicro25钢所受最大力(F)随压头下压时间(t)变化的模拟结果与实验结果对比。两种压头形状的模拟曲线均与实验结果呈现出较好的吻合趋势。为进一步定量评价模拟精度，表1统计了1500 s时间节点的作用力绝对误差，并采用标准差(standard deviation，STDEV)对全程曲线差异性进行表征。标准差的计算基于模拟值与实验值的偏差序列，其数学表达式为：

S T D E V = \sqrt[]{\frac{1}{n - 1}} \sum_{i = 1}^{n} {[(E_{i} - S_{i}) - μ]}^{2}

(1)

式中， $E_{i}$ 和 $S_{i}$ 分别为第i个数据点的实验值和模拟值； $μ$ 为偏差序列的均值；n为数据总量。分析结果表明，圆锥形压头在1500 s的作用力误差为2.20%，全程曲线STDEV达4.55，表明其模拟结果存在局部失稳；四棱锥形压头在1500 s的瞬时误差(1.68%)小于圆锥形压头，但STDEV显著升高至6.90。

图3

图3 Sanicro25钢在圆锥形压头和四棱锥形压头在最大压痕深度0.1 mm下的压痕模拟结果与实验结果对比

Fig.3 Comparison of simulation and experimental results for Sanicro25 steel under conical (a) and quadrangular pyramid (b) indenters with a depth of 0.1 mm (F—maximum force, t—compression head pressing time)

表1 Sanicro25钢在不同压头作用下的实验与模拟结果对比

Table 1 Comparison of experimental and simulation results of Sanicro25 steel under different indenters (The maximum penetration depth is 0.1 mm)

Indenter shape	Force at 1500 s / N		Error %	Curvature difference (STDEV)
Indenter shape	Experiment	Simulation	Error %	Curvature difference (STDEV)
Conical	276.36	270.28	2.20	4.55
Quadrangular pyramid	335.21	329.57	1.68	6.90

Note: STDEV—standard deviation

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综合误差与标准差指标可知，圆锥形压头的模拟结果具有更高的全局一致性，其拟合优度优于四棱锥形压头，该结论通过图3a中实验曲线与模拟曲线的紧密贴合得到进一步验证。

3 分析与讨论

3.1 网格密度依赖性分析

网格密度是影响模拟结果的一个重要因素^[25,26]。为了研究网格密度依赖性，同等程度地改变距压痕中心0.05 mm范围内3个方向的网格密度，共采用了11种网格尺寸，分别为0.004、0.008、0.012、0.015、0.020、0.025、0.030、0.035、0.040、0.045、0.050。在所有模拟工况中保持0.1 mm的最大压痕深度以确保结果的可比性。图4给出了被测试样模型网格密度对松弛曲线的影响。可以看出，网格密度对时间-载荷曲线的影响可以忽略不计，不同网格划分对模拟结果的精确性影响在可接受范围内。因此，为兼顾计算效率与结果精度，本工作选取最小网格长度位于尺寸序列中间值的0.025 mm建立有限元模型，以确保模拟的可靠性。

图4

图4 试样模型网格密度对松弛曲线的影响

Fig.4 Effect of mesh density of sample model on relaxation curve

3.2 厚深比对压痕松弛行为的影响

在压痕松弛过程中，试样厚度和压入深度作为关键参数显著影响测试结果的准确性，当试样厚度过薄且压入深度较大时，试验台基体效应的干扰会导致试样受力状态发生变化，从而引起实验数据的系统性偏差；反之若试样过厚，则难以充分发挥压痕实验所需测试材料量少的优势^[13]。ISO 14577-1：2015标准建议，试样厚度应至少为压痕深度的10倍(厚深比≥ 10)，以避免基底效应对测试结果的影响，但是该范围仅为一个基础性的参考值，因此合理选择试样厚度与压入深度的匹配关系，对于确保实验结果的可靠性及保持该方法的技术优势具有重要意义。模拟中采用0.1 mm的最大压痕深度，通过改变试样厚度来建立不同的厚深比关系，选取0.25、1、2、2.4、3、4、6、8和16 mm共9组试样厚度，对应的厚深比分别为2.5、10、20、24、30、40、60、80和160。

图5给出了厚深比对松弛曲线和最大力的影响结果。从图5a可以看出，厚深比对压痕的松弛曲线有较大的影响。当厚深比为2.5和10时，由于试样厚度较小，导致试验台基体效应增强，使得试样受力状态发生改变，相同压痕深度下的最大力随厚度减小而增大，松弛曲线呈现明显差异；当厚深比增至20及以上时，基体效应趋于稳定，各试样的最大力基本一致，松弛曲线基本重合。图5b的最大力-厚深比关系进一步证实，在厚深比小于20的范围内，最大力随厚深比增大呈现快速衰减趋势；当厚深比超过20后，载荷变化趋于平稳并近似收敛于稳定值。

图5

图5 厚深比对松弛曲线和最大力的影响

Fig.5 Effects of thickness-to-depth ratio on relaxation curve (a) and maximum force (b)

3.3 摩擦系数对压痕松弛行为的影响

在压痕有限元模拟中，为降低计算复杂度和提高计算效率，通常假设压头与试样接触界面为理想无摩擦状态，但在实际接触过程中摩擦效应是客观存在的，且会对压痕周围材料的堆积形貌产生一定影响。为系统研究摩擦系数对压痕松弛响应的影响规律，本工作在有限元模型中引入界面摩擦因素，设置0、0.10、0.15、0.20和0.30等5组不同摩擦系数进行对比分析。不同摩擦系数下松弛曲线的模拟结果如图6所示。可见，摩擦系数对最大力影响较大，其在0~0.30范围内变化时最大力的相对偏差最大可达18.41%。

图6

图6 摩擦系数对松弛曲线的影响

Fig.6 Effect of friction coefficient (f) on relaxation curve

根据图7a红色直线标记的路径提取压痕表面轮廓图，结果如图7b所示。可以看出，摩擦系数在0.00~0.30范围内，压痕周围材料的堆积行为有明显的摩擦依赖性：随着摩擦系数从0增至0.15，表面材料堆积高度持续降低；而当摩擦系数超过0.15后，继续增大摩擦系数，材料堆积高度不再进一步降低，即压痕表面材料堆积的减少逐渐趋于稳定。

图7

图7 不同摩擦系数下根据红色路径提取的压痕表面轮廓

Fig.7 From the red path (a) on the specimen surface, the indentation surface profile diagrams (b) extracted under different friction coefficients (Inset in Fig.7b is the locally enlarged diagram)

3.4 压头几何形状对压痕松弛行为的影响

3.4.1 压痕应力分布特征

为系统探究锥形压头几何形状对Sanicro25钢蠕变行为的影响，首先对圆锥和四棱锥形状压头作用下的应力场分布特征进行对比分析。图8和9分别为两种压头在最大压痕深度以及卸载后压痕变形区的等效应力分布云图。

图8

图8 圆锥形和四棱锥形压头在最大压痕深度(0.1 mm)下试样表面压痕Mises应力分布

Fig.8 Mises stress (S) distributions of indentation on specimen surface under maximum indentation (0.1 mm) depth with conical (a) and quadrangular pyramid (b) indenters

图9

图9 圆锥形和四棱锥形压头完全卸载后试样表面压痕Mises应力分布

Fig.9 Mises stress distributions of indentation on specimen surface after conical (a) and quadrangular pyramid (b) indenters are completely unloaded (SR—stress ring)

两种压头形状下的应力场呈现以下共同规律：在最大压痕深度时，等效应力极值均集中于压头尖端与材料接触的微区，表现出典型的接触应力集中现象^[27]；载荷卸载后的应力场均呈圆形，应力场的分布面积明显增大(面积较加载状态增大58.46%~76.70%)，并且所有压头工况下均观测到明显的残余应力环(stress ring)，证实了材料在蠕变过程中发生了塑性应变。

但是，不同形状压头作用下应力分布的差异更为显著：在相同压痕深度和蠕变时间下，压痕应力的大小因压头形状的不同而变化。圆锥和四棱锥形状压头作用下的最大等效应力依次升高，分别为1345.00和1728.00 MPa；载荷卸载后，最大等效应力明显降低，分别为1036.00和1531.00 MPa。圆锥形压头的应力释放率为22.97%，大于四棱锥形的11.40%，这种差异主要源于不同压头几何形状在变形过程中产生的约束效应差异，圆锥形压头由于几何约束较小而表现出更显著的应力释放特性，四棱锥形压头由于棱边应力集中则以塑性应变为主。此外，最大压痕深度时，圆锥与四棱锥形状压头作用下的压痕Mises应力云图分别呈圆形和正方形；完全卸载后，圆锥形、四棱锥形压头的残余应力环分别在86.32~172.60和127.60~255.20 MPa范围。

图10给出了圆锥形压痕中心剖面和四棱锥形沿压痕斜面中线剖面在压痕最大深度和完全卸载后的应力分布云图。分析表明，在最大压入深度时，等效应力呈现由压痕中心向四周递减的分布特征(图10a和c)；卸载后，圆锥压痕中心区、四棱锥压痕中心及正下方区域、对角线区域以及棱锥尖角处均存在明显的应力集中现象，而四棱锥斜面区域与圆锥环形区域的残余应力相对较小，如图10b和d所示。这种差异主要源于不同压头几何形状导致的塑性变形不均匀分布，其中四棱锥的棱边效应和圆锥的轴对称特性对残余应力分布模式产生了决定性影响。

图10

图10 圆锥形压头和四棱锥形压头作用下Sanicro25钢在压痕最大深度和完全卸载后的剖面Mises应力分布云图(最大压深0.1 mm)

Fig.10 Profile Mises stress distribution nephograms and partially enlarged views (insets) of Sanicro25 steel at maximum inden-tation depth (a, c) and after unloading (b, d) under two shapes of indenters (The maximum penetration depth is 0.1 mm)

(a, b) conical indenter (c, d) quadrangular pyramid indenter

在如图11a和c所示的四分之一模型上创建节点路径并分析完全卸载后的残余应力分布特征，结果如图11b和d所示。其中，图11b和d中的插图是其局部放大图，路径1沿压痕中心Z轴向下，路径2沿X轴水平延伸，针对四棱锥压头增设路径3沿对角线向外。

图11

图11 圆锥形压头和四棱锥形压头作用下Sanicro25钢在压头完全卸载后沿不同路径的残余应力分布曲线

Fig.11 After complete unloading of the conical indenter (a, b) and quadrangular pyramidal indenter (c, d), residual stress distribution curves (b, d) of Sanicro25 steel along different paths (a, c) on the specimen are shown (Insets in Figs.11b and d are the locally enlarged diagrams)

从整体趋势上看，最大残余应力位于压痕中心区域。其中，圆锥形和四棱锥形压头作用下路径1上的残余应力沿Z轴方向均呈现出“先降低-再升高-再降低”的变化规律，残余应力在压痕中心区的正下方发生了应力集中现象，应力分别在59.59~118.42和64.23~90.88 MPa之间。路径2上的残余应力分布较为均匀，分别在17.82~87.95和18.98~81.14 MPa之间，但残余应力在压痕变形区的“边界”均发生不同程度的应力集中，应力分别在116.27~197.62和115.15~238.81 MPa之间。四棱锥形压头作用下路径3上的残余应力在压痕对角线区域以及尖角区域发生了较大程度的应力集中，应力范围为444.35~1456.28 MPa。

3.4.2 压痕等效应变分布

当前增量步的等效塑性应变(PEEQ)是描述材料塑性变形程度的标量，定义为：

P E E Q = \int_{0}^{l} \dot{\bar{ε^{p l}}} d t = \int \sqrt[]{\frac{2}{3} ({\dot{ε}}_{i j}^{p l})^{2}} d t

(2)

式中， $\dot{\bar{ε^{p l}}}$ 为等效塑性应变率， ${\dot{ε}}_{i j}^{p l}$ 为塑性应变率张量的分量，l为距压痕中心的距离。

等效蠕变应变(CEEQ)是等效蠕变应变率( $\dot{\bar{ε^{c r}}}$ )在一定时间段内的积分，常用其表示材料的蠕变变形行为^[28]，定义为：

C E E Q = \int_{0}^{l} \dot{\bar{ε^{c r}}} d t = \int \sqrt[]{\frac{2}{3} ({\dot{ε}}_{i j}^{c r})^{2}} d t

(3)

式中， ${\dot{ε}}_{i j}^{c r}$ 为蠕变应变率张量的分量。

图12展示了两种形状压头作用下Sanicro25钢在压痕最大深度和完全卸载后试件表面的等效塑性应变和等效蠕变应变云图。可以看出，压痕最大深度条件下，圆锥、四棱锥形状压头作用下的最大等效塑性应变(图12a和b)和最大等效蠕变应变(图12c和d)也都发生在压头尖端与被测材料相接触的微小区域，与应力云图对应分别呈圆形和正方形，但是圆锥形压头压入时变形更对称，四棱锥压头在棱边处则易诱发局部塑性变形。圆锥、四棱锥压头作用下的最大等效塑性应变分别为0.5258和0.5268，相差不大；但四棱锥压头会产生比圆锥压头更大的最大等效蠕变应变，分别为0.02153和0.01465。

图12

图12 圆锥形压头和四棱锥形压头作用下Sanicro25钢在压痕最大深度(0.1 mm)和完全卸载后表面的等效塑性应变和等效蠕变应变云图

Fig.12 Equivalent plastic strain (PEEQ) (a, b, e, f) and equivalent creep strain (CEEQ) (c, d, g, h) nephograms of Sanicro25 steel under two shapes of indenters at the maximum indentation depth (0.1 mm) (a-d) and after complete unloading (e-h) on the surface

其次，圆锥形状压头作用下塑性应变和蠕变应变主要发生在压痕中心区域，取值范围分别为0.482~0.526和0.013~0.015；四棱锥形状压头作用下塑性应变和蠕变应变主要发生在压痕对角线区域，取值范围分别为0.483~0.527和0.020~0.022，4个斜面区域的塑性应变和蠕变应变则维持在较低水平，取值范围分别为0~0.044和0~0.002，这表明压痕斜面区域的变形受压头的影响较压痕对角线小。在远离压痕变形区的其他区域，Sanicro25钢未发生明显的塑性变形和蠕变变形。

此外，完全卸载后Sanicro25钢的弹性恢复行为导致压痕变形区的等效应变分布范围都有所增大。在等效塑性应变方面(图12e和f)，不同压头形状下压痕变形区的最大值均保持相对稳定，未发生明显变化；在等效蠕变应变方面(图12g和h)，四棱锥压头作用区域的最大值稍有降低，而圆锥压头作用区域则表现出相反的变化规律，其最大值略有升高。

图13为Sanicro25钢分别在圆锥形和四棱锥形压头卸载后沿不同路径的等效应变分布曲线。结果表明，不同压头形状作用下，在被测材料的边缘，等效塑性应变和等效蠕变应变均近似为0，即可认为远离压痕的区域不发生变形。圆锥形压头作用下，在远离压痕变形中心区的过程中，等效塑性应变随之降低，而四棱锥形压头作用下的等效塑性应变在棱边区域集中；圆锥形压头作用下的最大等效蠕变应变发生在压痕变形中心区的正下方，而四棱锥形压头作用下的最大等效蠕变应变则发生在棱边尖端区域。

图13

图13 Sanicro25钢在两种形状压头完全卸载后沿不同路径的等效塑性应变和等效蠕变应变分布曲线

Fig.13 Along different paths (a, d) on the specimen of Sanicro25 steel after unloading by two shapes of indenters, PEEQ (b, e) and CEEQ (c, f) distribution curves are shown (Insets in Figs.13c, e, and f are the locally enlarged diagrams)

(a-c) conical indenter (d-f) quadrangular pyramid indenter

3.4.3 不同形状压头下的松弛曲线

图14为Sanicro25钢在圆锥形和四棱锥形压头作用下，0.1 mm恒定压入深度条件下的松弛曲线。由图可知，不同压头作用下的松弛曲线呈现相同的趋势，可分成两个阶段：第一阶段最大力值随时间迅速减小，松弛速率逐渐降低，这主要是由于压头前方受压材料应力状态调整所致；第二阶段为“稳定”阶段，最大力值以极小的速率缓慢减小，松弛速率逐渐达到恒定^[28]。

图14

图14 两种压头在0.1 mm最大压深下的松弛曲线

Fig.14 Relaxation curves of two indenters at 0.1 mm maximum pressure depth

由前文应力分布结果可知，四棱锥棱边处存在应力集中，这导致四棱锥压头作用下的初始松弛速率快于圆锥压头，并且为了达到相同的位移，圆锥压头施加的力应大于四棱锥压头施加的力。

4 结论

(1) 在750 ℃、最大压痕深度为0.1 mm条件下，被测试样模型的最小网格长度在0.004~0.050范围内变化对压痕松弛曲线影响很小，距压痕中心0.05 mm范围内选择最小网格长度为0.025的有限元模型是可靠的。

(2) 摩擦系数对测试结果有明显影响，当其值从0增加到0.30时，对应的最大力增大了18.41%；同时会明显改变压痕表面形貌，随着摩擦系数的增大，材料堆积高度递减，但当摩擦系数超过0.15后，堆积高度的变化幅度显著减小，不同摩擦系数下的结果趋于一致。

(3) 当厚深比较小(< 20)时，由于试样厚度不足引起的边界效应会导致模拟结果产生显著偏差；当该比值≥ 20时，松弛曲线基本一致，进一步增大厚深比不会引起曲线的显著变化。在实验中应将厚深比控制在20以上以确保数据的可靠性。

(4) 压头几何形状显著影响应力场的分布特征。圆锥形压头产生典型的轴对称应力场；四棱锥形压头由于棱边效应导致应力集中现象明显，会加速初始松弛速率。对于给定尺寸的两种压头，在相同压痕深度作用下，四棱锥压头会产生比圆锥压头更大的等效蠕变应变。

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